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Pythagorean Theorem and Its Converse in Right Triangles, Lecture notes of Construction

A proof of the Pythagorean Theorem and its converse in the context of right triangles using given sides and hypotenuse lengths. It also includes examples and exercises.

What you will learn

  • How can you prove the Pythagorean Theorem for a right triangle?
  • How can you use the Pythagorean Theorem to determine if a triangle is a right triangle?
  • What is the Pythagorean Theorem in the context of right triangles?
  • What is the converse of the Pythagorean Theorem in the context of right triangles?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Find x.
1.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 13
and the lengths of the legs are 5 and x.
ANSWER:
12
2.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The length of the hypotenuse is x and the lengths of
thelegsare8and12.
ANSWER:
3.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 16
and the lengths of the legs are 4 and x.
ANSWER:
4.Use a Pythagorean triple to find x. Explain your
reasoning.
SOLUTION:
35 is the hypotenuse, so it is the greatest value in the
Pythagorean Triple.
Find the common factors of 35 and 21.
TheGCFof35and21is7.Dividethisout.
Check to see if 5 and 3 are part of a Pythagorean
triplewith5asthelargestvalue.
We have one Pythagorean triple 3-4-5. The multiples
of this triple also will be Pythagorean triple. So, x = 7
(4) = 28.
Use the Pythagorean Theorem to check it.
ANSWER:
28; Since and and 3-4-5 is a
Pythagorean triple, or 28.
5.MULTIPLE CHOICE The mainsail of a boat is
shown. What is the length, in feet, of ?
A 52.5
B 65
C 72.5
D 75
SOLUTION:
The mainsail is in the form of a right triangle. The
length of the hypotenuse is x and the lengths of the
legsare45and60.
Therefore, the correct choice is D.
ANSWER:
D
Determine whether each set of numbers can be
the measures of the sides of a triangle. If so,
classify the triangle as acute, obtuse, or right.
Justify your answer.
6.15, 36, 39
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
Yes; right
7.16, 18, 26
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
8.15, 20, 24
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an acute
triangle.
ANSWER:
Yes; acute
Find x.
9.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare12and16.
ANSWER:
20
10.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 15
and the lengths of the legs are 9 and x.
ANSWER:
12
11.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 5
and the lengths of the legs are 2 and x.
ANSWER:
12.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 66
and the lengths of the legs are 33 and x.
ANSWER:
13.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are .
ANSWER:
14.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is
andthelengthsofthelegsare .
ANSWER:
CCSSPERSEVERANCEUseaPythagorean
Triple to find x.
15.
SOLUTION:
Find the greatest common factors of 16 and 30.
The GCF of 16 and 30 is 2. Divide this value out.
Check to see if 8 and 15 are part of a Pythagorean
triple.
We have one Pythagorean triple 8-15-17. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(17) = 34.
Use the Pythagorean Theorem to check it.
ANSWER:
34
16.
SOLUTION:
Findthegreatestcommonfactorof14and48.
The GCF of 14 and 48 is 2. Divide this value out.
Check to see if 7 and 24 are part of a Pythagorean
triple.
We have one Pythagorean triple 7-24-25. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(25) = 50.
Use the Pythagorean Theorem to check it.
ANSWER:
50
17.
SOLUTION:
74 is the hypotenuse, so it is the greatest value in the
Pythagorean triple.
Findthegreatestcommonfactorof74and24.
The GCF of 24 and 74 is 2. Divide this value out.
Check to see if 12 and 37 are part of a Pythagorean
triplewith37asthelargestvalue.
We have one Pythagorean triple 12-35-37. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(35) = 70.
Use the Pythagorean Theorem to check it.
ANSWER:
70
18.
SOLUTION:
78 is the hypotenuse, so it is the greatest value in the
Pythagorean triple.
Find the greatest common factor of 78 and 72.
78=2×3×13
72=2×2×2×3×3
The GCF of 78 and 72 is 6. Divide this value out.
78÷6=13
72÷6=12
Check to see if 13 and 12 are part of a Pythagorean
triple with 13 as the largest value.
132 122 = 169 144 = 25 = 52
We have one Pythagorean triple 5-12-13. The
multiples of this triple also will be Pythagorean triple.
So, x = 6(5) = 30.
Use the Pythagorean Theorem to check it.
ANSWER:
30
19.BASKETBALL The support for a basketball goal
forms a right triangle as shown. What is the length x
of the horizontal portion of the support?
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is
andthelengthsofthelegsare
.
Therefore, the horizontal position of the support is
about 3 ft.
ANSWER:
about 3 ft
20.DRIVING The street that Khaliah usually uses to
get to school is under construction. She has been
taking the detour shown. If the construction starts at
the point where Khaliah leaves her normal route and
ends at the point where she re-enters her normal
route, about how long is the stretch of road under
construction?
SOLUTION:
Let x be the length of the road that is under
construction. The road under construction and the
detour form a right triangle.
The length of the hypotenuse is x and the lengths of
thelegsare0.8and1.8.
Therefore, a stretch of about 2 miles is under
construction.
ANSWER:
about 2 mi
Determine whether each set of numbers can be
the measures of the sides of a triangle. If so,
classify the triangle as acute, obtuse, or right.
Justify your answer.
21.7, 15, 21
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
22.10, 12, 23
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Since the sum of the lengths of two sides is less than
that of the third side, the set of numbers cannot be
measures of a triangle.
ANSWER:
No; 23 > 10 + 12
23.4.5, 20, 20.5
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
Yes; right
24.44, 46, 91
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Since the sum of the lengths of two sides is less than
that of the third side, the set of numbers cannot be
measures of a triangle.
ANSWER:
No; 91 > 44 + 46
25.4.2, 6.4, 7.6
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an acute
triangle.
ANSWER:
Yes; acute
26.4 , 12, 14
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
Find x.
27.
SOLUTION:
The triangle with the side lengths 9, 12, and x form a
right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare9and12.
ANSWER:
15
28.
SOLUTION:
The segment of length 16 units is divided to two
congruent segments. So, the length of each segment
will be 8 units. Then we have a right triangle with the
sides 15, 8, and x.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare8and15.
ANSWER:
17
29.
SOLUTION:
We have a right triangle with the sides 14, 10, and x.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 14
and the lengths of the legs are 10 and x.
ANSWER:
COORDINATE GEOMETRY Determine
whether is an acute, right, or obtuse
triangle for the given vertices. Explain.
30.X(3, 2), Y(1, 0), Z(0, 1)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
right; , ,
31.X(7, 3), Y(2, 5), Z(4, 1)
SOLUTION:
Use the distance formula to find the length of each si
triangle.
Classify the triangle by comparing the square of the l
the sum of the squares of the other two sides.
Therefore, by the Pythagorean Inequality Theorem, a
the given measures will be an acute triangle.
ANSWER:
acute; , ,
;
32.X(1, 2), Y(4, 6), Z(6, 6)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
obtuse; XY = 5, YZ = 2, ;
33.X(3, 1), Y(3, 7), Z(11, 1)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
right; XY = 6, YZ = 10, XZ = 8; 62 + 82 = 102
34.JOGGING Brett jogs in the park three times a
week. Usually, he takes a -mile path that cuts
through the park. Today, the path is closed, so he is
taking the orange route shown. How much farther
will he jog on his alternate route than he would have
if he had followed his normal path?
SOLUTION:
The normal jogging route and the detour form a right
triangle. One leg of the right triangle is 0.45 mi. and
let x be the other leg. The hypotenuse of the right
triangle is . In a right triangle, the sum of the
squares of the lengths of the legs is equal to the
square of the length of the hypotenuse.
So, the total distance that he runs in the alternate
route is 0.45 + 0. 6 = 1.05 mi. instead of his normal
distance 0.75 mi.
Therefore, he will be jogging an extra distance of 0.3
miles in his alternate route.
ANSWER:
0.3 mi
35.PROOF Write a paragraph proof of Theorem 8.5.
SOLUTION:
Theorem 8.5 states that if the sum of the squares of
the lengths of the shortest sides of a triangle are
equal to the square of the length of the longest side,
then the triangle is a right triangle. Use the
hypothesis to create a given statement, namely that
c2 = a2 + b2
fortriangleABC.Youcanaccomplishthisproofby
constructing another triangle (Triangle DEF) that is
congruent to triangle ABC, using SSS triangle
congruence theorem. Show that triangle DEF is a
right triangle, using the Pythagorean theorem.
therefore, any triangle congruent to a right triangle
butalsobearighttriangle.
Given: with sides of measure a, b, and c,
where c2 = a2 + b2
Prove: is a right triangle.
Proof: Draw on line with measure equal to a.
At D, draw line . Locate point F on m so that
DF = b. Draw and call its measure x.
Because is a right triangle, a2 + b2 = x2. But
a2 + b2 = c2, so x2 = c2 or x = c. Thus,
by SSS. This means .
Therefore, C must be a right angle, making
a right triangle.
ANSWER:
Given: with sides of measure a, b, and c,
where c2 = a2 + b2
Prove: is a right triangle.
Proof: Draw on line with measure equal to a.
At D, draw line . Locate point F on m so that
DF = b. Draw and call its measure x.
Because is a right triangle, a2 + b2 = x2. But
a2 + b2 = c2, so x2 = c2 or x = c. Thus,
by SSS. This means .
Therefore, C must be a right angle, making
a right triangle.
PROOF Write a two-column proof for each
theorem.
36.Theorem 8.6
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two triangles, a right
angle, relationship between sides. Use the properties
that you have learned about right angles, acute
angles, Pythagorean Theorem,angle relationships and
equivalent expressions in algebra to walk through the
proof.
Given: In , c2 < a2 + b2 where c is the length
of the longest side. In , R is a right angle.
Prove: is an acute triangle.
Proof:
Statements (Reasons)
1. In , c2 < a2 + b2 where c is the length of
the longest
side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 < x2 (Substitution Property)
4. c < x (A property of square roots)
5. m R = (Definition of a right angle)
6. m C < m R (Converse of the Hinge Theorem)
7. m C < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofan
acute triangle)
ANSWER:
Given: In , c2 < a2 + b2 where c is the length
of the longest side. In , R is a right angle.
Prove: is an acute triangle.
Proof:
Statements (Reasons)
1. In , c2 < a2 + b2 where c is the length of
the longest
side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 < x2 (Substitution Property)
4. c < x (A property of square roots)
5. m R = (Definition of a right angle)
6. m C < m R (Converse of the Hinge Theorem)
7. m C < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofanacute
triangle)
37.Theorem 8.7
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two triangles, relationship
between angles.Use the properties that you have
learned about triangles, angles and equivalent
expressions in algebra to walk through the proof.
Given: In , c2 > a2 + b2, where c is the length
of the longest side.
Prove: is an obtuse triangle.
Statements (Reasons)
1. In , c2 > a2 + b2, where c is the length of
the
longest side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 > x2 (Substitution Property)
4. c > x (A property of square roots)
5. m R = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
ANSWER:
Given: In , c2 > a2 + b2, where c is the length
of the longest side.
Prove: is an obtuse triangle.
Statements (Reasons)
1. In , c2 > a2 + b2, where c is the length of
the
longest side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 > x2 (Substitution Property)
4. c > x (A property of square roots)
5. m R = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
CCSS SENSE-MAKINGFindtheperimeter
and area of each figure.
38.
SOLUTION:
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. Since the triangle is a right triangle the
base and the height are the legs of the triangle. So,
The perimeter is the sum of the lengths of the three
sides. Use the Pythagorean Theorem to find the
length of the hypotenuse of the triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The length of the hypotenuse is 20 units. Therefore,
the perimeter is 12 + 16 + 20 = 48 units.
ANSWER:
P = 48 units; A = 96 units2
39.
SOLUTION:
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. The altitude to the base of an isosceles
triangle bisects the base. So, we have two right
triangles with one of the legs equal to 5 units and the
hypotenuse is 13 units each. Use the Pythagorean
Theorem to find the length of the common leg of the
triangles.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The altitude is 12 units.
Therefore,
The perimeter is the sum of the lengths of the three
sides. Therefore, the perimeter is 13 + 13 + 10 = 36
units.
ANSWER:
P = 36 units; A = 60 units2
40.
SOLUTION:
The given figure can be divided as a right triangle
and a rectangle as shown.
The total are of the figure is the sum of the areas of
the right triangle and the rectangle.
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. Since the triangle is a right triangle the
base and the height are the legs of the triangle. So,
The area of a rectangle of length l and width w is
given by the formula A = l×w. So,
Therefore, the total area is 24 + 32 = 56 sq. units.
The perimeter is the sum of the lengths of the four
boundaries. Use the Pythagorean Theorem to find
the length of the hypotenuse of the triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Thehypotenuseis10units.
Therefore, the perimeter is 4 + 8 + 10 + 10 = 32 units
ANSWER:
P = 32 units; A = 56 units2
41.ALGEBRA The sides of a triangle have lengths x, x
+ 5, and 25. If the length of the longest side is 25,
what value of x makes the triangle a right triangle?
SOLUTION:
By the converse of the Pythagorean Theorem, if the
square of the longest side of a triangle is the sum of
squares of the other two sides then the triangle is a
right triangle.
Use the Quadratic Formula to find the roots of the
equation.
Since x is a length, it cannot be negative. Therefore,
x = 15.
ANSWER:
15
42.ALGEBRA The sides of a triangle have lengths 2x,
8, and 12. If the length of the longest side is 2x, what
values of x make the triangle acute?
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
So, the value of x should be between 2 and 10.
By the Pythagorean Inequality Theorem, if the
square of the longest side of a triangle is less than the
sum of squares of the other two sides then the
triangleisanacutetriangle.
Therefore, for the triangle to be acute,
ANSWER:
43.TELEVISION The screen aspect ratio, or the ratio
of the width to the length, of a high-definition
television is 16:9. The size of a television is given by
the diagonal distance across the screen. If an HDTV
is 41 inches wide, what is its screen size?
Refer to the photo on page 547.
SOLUTION:
Use the ratio to find the length of the television.
Let x be the length of the television. Then,
Solve the proportion for x.
The two adjacent sides and the diagonal of the
television form a right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Therefore, the screen size is about 47 inches.
ANSWER:
47 in.
44.PLAYGROUND According to the Handbook for
Public Playground Safety, the ratio of the vertical
distance to the horizontal distance covered by a slide
should not be more than about 4 to 7. If the horizontal
distance allotted in a slide design is 14 feet,
approximately how long should the slide be?
SOLUTION:
Use the ratio to find the vertical distance.
Let x be the vertical distance. Then,
Solve the proportion for x.
The vertical distance, the horizontal distance, and the
slide form a right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Therefore, the slide will be about 16 ft long.
ANSWER:
about 16 ft
Find x.
45.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are x 4and8.
Solve for x.
ANSWER:
10
46.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are x 3and9.
Solve for x.
ANSWER:
15
47.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
+ 1 and the lengths of the legs are x and .
Solve for x.
ANSWER:
48.MULTIPLE REPRESENTATIONS In this
problem, you will investigate special right triangles.
a. GEOMETRIC Draw three different isosceles
right triangles that have whole-number side lengths.
Label the triangles ABC, MNP, and XYZ with the
right angle located at vertex A, M, and X,
respectively. Label the leg lengths of each side and
find the length of the hypotenuse in simplest radical
form.
b. TABULAR Copy and complete the table below.
c. VERBAL Make a conjecture about the ratio of
the hypotenuse to a leg of an isosceles right triangle.
SOLUTION:
a.Let AB = AC. Use the Pythagorean Theorem to
find BC.
Sampleanswer:
 
Let MN = PM. Use the Pythagorean Theorem to
find NP.

Let ZX = XY. Use the Pythagorean Theorem to find
ZY.

b.Complete the table below with the side lengths
calculated for each triangle. Then, compute the given
ratios.
c. Summarize any observations made based on the
patternsinthetable.
Sample answer: The ratio of the hypotenuse to a leg
of an isosceles right triangle is .
ANSWER:
a.
b.
c. Sample answer: The ratio of the hypotenuse to a
leg of an isosceles right triangle is .
49.CHALLENGE Find the value of x in the figure.
Find x.
1.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 13
and the lengths of the legs are 5 and x.
ANSWER:
12
2.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The length of the hypotenuse is x and the lengths of
thelegsare8and12.
ANSWER:
3.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 16
and the lengths of the legs are 4 and x.
ANSWER:
4.Use a Pythagorean triple to find x. Explain your
reasoning.
SOLUTION:
35 is the hypotenuse, so it is the greatest value in the
Pythagorean Triple.
Find the common factors of 35 and 21.
TheGCFof35and21is7.Dividethisout.
Check to see if 5 and 3 are part of a Pythagorean
triplewith5asthelargestvalue.
We have one Pythagorean triple 3-4-5. The multiples
of this triple also will be Pythagorean triple. So, x = 7
(4) = 28.
Use the Pythagorean Theorem to check it.
ANSWER:
28; Since and and 3-4-5 is a
Pythagorean triple, or 28.
5.MULTIPLE CHOICE The mainsail of a boat is
shown. What is the length, in feet, of ?
A 52.5
B 65
C 72.5
D 75
SOLUTION:
The mainsail is in the form of a right triangle. The
length of the hypotenuse is x and the lengths of the
legsare45and60.
Therefore, the correct choice is D.
ANSWER:
D
Determine whether each set of numbers can be
the measures of the sides of a triangle. If so,
classify the triangle as acute, obtuse, or right.
Justify your answer.
6.15, 36, 39
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
Yes; right
7.16, 18, 26
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
8.15, 20, 24
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an acute
triangle.
ANSWER:
Yes; acute
Find x.
9.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare12and16.
ANSWER:
20
10.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 15
and the lengths of the legs are 9 and x.
ANSWER:
12
11.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 5
and the lengths of the legs are 2 and x.
ANSWER:
12.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 66
and the lengths of the legs are 33 and x.
ANSWER:
13.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are .
ANSWER:
14.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is
andthelengthsofthelegsare .
ANSWER:
CCSSPERSEVERANCEUseaPythagorean
Triple to find x.
15.
SOLUTION:
Find the greatest common factors of 16 and 30.
The GCF of 16 and 30 is 2. Divide this value out.
Check to see if 8 and 15 are part of a Pythagorean
triple.
We have one Pythagorean triple 8-15-17. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(17) = 34.
Use the Pythagorean Theorem to check it.
ANSWER:
34
16.
SOLUTION:
Findthegreatestcommonfactorof14and48.
The GCF of 14 and 48 is 2. Divide this value out.
Check to see if 7 and 24 are part of a Pythagorean
triple.
We have one Pythagorean triple 7-24-25. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(25) = 50.
Use the Pythagorean Theorem to check it.
ANSWER:
50
17.
SOLUTION:
74 is the hypotenuse, so it is the greatest value in the
Pythagorean triple.
Findthegreatestcommonfactorof74and24.
The GCF of 24 and 74 is 2. Divide this value out.
Check to see if 12 and 37 are part of a Pythagorean
triplewith37asthelargestvalue.
We have one Pythagorean triple 12-35-37. The
multiples of this triple also will be Pythagorean triple.
So, x = 2(35) = 70.
Use the Pythagorean Theorem to check it.
ANSWER:
70
18.
SOLUTION:
78 is the hypotenuse, so it is the greatest value in the
Pythagorean triple.
Find the greatest common factor of 78 and 72.
78=2×3×13
72=2×2×2×3×3
The GCF of 78 and 72 is 6. Divide this value out.
78÷6=13
72÷6=12
Check to see if 13 and 12 are part of a Pythagorean
triple with 13 as the largest value.
132 122 = 169 144 = 25 = 52
We have one Pythagorean triple 5-12-13. The
multiples of this triple also will be Pythagorean triple.
So, x = 6(5) = 30.
Use the Pythagorean Theorem to check it.
ANSWER:
30
19.BASKETBALL The support for a basketball goal
forms a right triangle as shown. What is the length x
of the horizontal portion of the support?
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is
andthelengthsofthelegsare
.
Therefore, the horizontal position of the support is
about 3 ft.
ANSWER:
about 3 ft
20.DRIVING The street that Khaliah usually uses to
get to school is under construction. She has been
taking the detour shown. If the construction starts at
the point where Khaliah leaves her normal route and
ends at the point where she re-enters her normal
route, about how long is the stretch of road under
construction?
SOLUTION:
Let x be the length of the road that is under
construction. The road under construction and the
detour form a right triangle.
The length of the hypotenuse is x and the lengths of
thelegsare0.8and1.8.
Therefore, a stretch of about 2 miles is under
construction.
ANSWER:
about 2 mi
Determine whether each set of numbers can be
the measures of the sides of a triangle. If so,
classify the triangle as acute, obtuse, or right.
Justify your answer.
21.7, 15, 21
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
22.10, 12, 23
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Since the sum of the lengths of two sides is less than
that of the third side, the set of numbers cannot be
measures of a triangle.
ANSWER:
No; 23 > 10 + 12
23.4.5, 20, 20.5
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
Yes; right
24.44, 46, 91
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Since the sum of the lengths of two sides is less than
that of the third side, the set of numbers cannot be
measures of a triangle.
ANSWER:
No; 91 > 44 + 46
25.4.2, 6.4, 7.6
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an acute
triangle.
ANSWER:
Yes; acute
26.4 , 12, 14
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
Therefore, the set of numbers can be measures of a
triangle.
Now, classify the triangle by comparing the square of
the longest side to the sum of the squares of the
other two sides.
Therefore, by Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
Yes; obtuse
Find x.
27.
SOLUTION:
The triangle with the side lengths 9, 12, and x form a
right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare9and12.
ANSWER:
15
28.
SOLUTION:
The segment of length 16 units is divided to two
congruent segments. So, the length of each segment
will be 8 units. Then we have a right triangle with the
sides 15, 8, and x.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
andthelengthsofthelegsare8and15.
ANSWER:
17
29.
SOLUTION:
We have a right triangle with the sides 14, 10, and x.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is 14
and the lengths of the legs are 10 and x.
ANSWER:
COORDINATE GEOMETRY Determine
whether is an acute, right, or obtuse
triangle for the given vertices. Explain.
30.X(3, 2), Y(1, 0), Z(0, 1)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
right; , ,
31.X(7, 3), Y(2, 5), Z(4, 1)
SOLUTION:
Use the distance formula to find the length of each si
triangle.
Classify the triangle by comparing the square of the l
the sum of the squares of the other two sides.
Therefore, by the Pythagorean Inequality Theorem, a
the given measures will be an acute triangle.
ANSWER:
acute; , ,
;
32.X(1, 2), Y(4, 6), Z(6, 6)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the Pythagorean Inequality Theorem, a
triangle with the given measures will be an obtuse
triangle.
ANSWER:
obtuse; XY = 5, YZ = 2, ;
33.X(3, 1), Y(3, 7), Z(11, 1)
SOLUTION:
Use the distance formula to find the length of each
side of the triangle.
Classify the triangle by comparing the square of the
longest side to the sum of the squares of the other
two sides.
Therefore, by the converse of Pythagorean Theorem,
a triangle with the given measures will be a right
triangle.
ANSWER:
right; XY = 6, YZ = 10, XZ = 8; 62 + 82 = 102
34.JOGGING Brett jogs in the park three times a
week. Usually, he takes a -mile path that cuts
through the park. Today, the path is closed, so he is
taking the orange route shown. How much farther
will he jog on his alternate route than he would have
if he had followed his normal path?
SOLUTION:
The normal jogging route and the detour form a right
triangle. One leg of the right triangle is 0.45 mi. and
let x be the other leg. The hypotenuse of the right
triangle is . In a right triangle, the sum of the
squares of the lengths of the legs is equal to the
square of the length of the hypotenuse.
So, the total distance that he runs in the alternate
route is 0.45 + 0. 6 = 1.05 mi. instead of his normal
distance 0.75 mi.
Therefore, he will be jogging an extra distance of 0.3
miles in his alternate route.
ANSWER:
0.3 mi
35.PROOF Write a paragraph proof of Theorem 8.5.
SOLUTION:
Theorem 8.5 states that if the sum of the squares of
the lengths of the shortest sides of a triangle are
equal to the square of the length of the longest side,
then the triangle is a right triangle. Use the
hypothesis to create a given statement, namely that
c2 = a2 + b2
fortriangleABC.Youcanaccomplishthisproofby
constructing another triangle (Triangle DEF) that is
congruent to triangle ABC, using SSS triangle
congruence theorem. Show that triangle DEF is a
right triangle, using the Pythagorean theorem.
therefore, any triangle congruent to a right triangle
butalsobearighttriangle.
Given: with sides of measure a, b, and c,
where c2 = a2 + b2
Prove: is a right triangle.
Proof: Draw on line with measure equal to a.
At D, draw line . Locate point F on m so that
DF = b. Draw and call its measure x.
Because is a right triangle, a2 + b2 = x2. But
a2 + b2 = c2, so x2 = c2 or x = c. Thus,
by SSS. This means .
Therefore, C must be a right angle, making
a right triangle.
ANSWER:
Given: with sides of measure a, b, and c,
where c2 = a2 + b2
Prove: is a right triangle.
Proof: Draw on line with measure equal to a.
At D, draw line . Locate point F on m so that
DF = b. Draw and call its measure x.
Because is a right triangle, a2 + b2 = x2. But
a2 + b2 = c2, so x2 = c2 or x = c. Thus,
by SSS. This means .
Therefore, C must be a right angle, making
a right triangle.
PROOF Write a two-column proof for each
theorem.
36.Theorem 8.6
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two triangles, a right
angle, relationship between sides. Use the properties
that you have learned about right angles, acute
angles, Pythagorean Theorem,angle relationships and
equivalent expressions in algebra to walk through the
proof.
Given: In , c2 < a2 + b2 where c is the length
of the longest side. In , R is a right angle.
Prove: is an acute triangle.
Proof:
Statements (Reasons)
1. In , c2 < a2 + b2 where c is the length of
the longest
side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 < x2 (Substitution Property)
4. c < x (A property of square roots)
5. m R = (Definition of a right angle)
6. m C < m R (Converse of the Hinge Theorem)
7. m C < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofan
acute triangle)
ANSWER:
Given: In , c2 < a2 + b2 where c is the length
of the longest side. In , R is a right angle.
Prove: is an acute triangle.
Proof:
Statements (Reasons)
1. In , c2 < a2 + b2 where c is the length of
the longest
side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 < x2 (Substitution Property)
4. c < x (A property of square roots)
5. m R = (Definition of a right angle)
6. m C < m R (Converse of the Hinge Theorem)
7. m C < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofanacute
triangle)
37.Theorem 8.7
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two triangles, relationship
between angles.Use the properties that you have
learned about triangles, angles and equivalent
expressions in algebra to walk through the proof.
Given: In , c2 > a2 + b2, where c is the length
of the longest side.
Prove: is an obtuse triangle.
Statements (Reasons)
1. In , c2 > a2 + b2, where c is the length of
the
longest side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 > x2 (Substitution Property)
4. c > x (A property of square roots)
5. m R = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
ANSWER:
Given: In , c2 > a2 + b2, where c is the length
of the longest side.
Prove: is an obtuse triangle.
Statements (Reasons)
1. In , c2 > a2 + b2, where c is the length of
the
longest side. In , R is a right angle. (Given)
2. a2 + b2 = x2 (Pythagorean Theorem)
3. c2 > x2 (Substitution Property)
4. c > x (A property of square roots)
5. m R = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
CCSS SENSE-MAKINGFindtheperimeter
and area of each figure.
38.
SOLUTION:
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. Since the triangle is a right triangle the
base and the height are the legs of the triangle. So,
The perimeter is the sum of the lengths of the three
sides. Use the Pythagorean Theorem to find the
length of the hypotenuse of the triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The length of the hypotenuse is 20 units. Therefore,
the perimeter is 12 + 16 + 20 = 48 units.
ANSWER:
P = 48 units; A = 96 units2
39.
SOLUTION:
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. The altitude to the base of an isosceles
triangle bisects the base. So, we have two right
triangles with one of the legs equal to 5 units and the
hypotenuse is 13 units each. Use the Pythagorean
Theorem to find the length of the common leg of the
triangles.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
The altitude is 12 units.
Therefore,
The perimeter is the sum of the lengths of the three
sides. Therefore, the perimeter is 13 + 13 + 10 = 36
units.
ANSWER:
P = 36 units; A = 60 units2
40.
SOLUTION:
The given figure can be divided as a right triangle
and a rectangle as shown.
The total are of the figure is the sum of the areas of
the right triangle and the rectangle.
The area of a triangle is given by the formula
whereb is the base and h is the height of
the triangle. Since the triangle is a right triangle the
base and the height are the legs of the triangle. So,
The area of a rectangle of length l and width w is
given by the formula A = l×w. So,
Therefore, the total area is 24 + 32 = 56 sq. units.
The perimeter is the sum of the lengths of the four
boundaries. Use the Pythagorean Theorem to find
the length of the hypotenuse of the triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Thehypotenuseis10units.
Therefore, the perimeter is 4 + 8 + 10 + 10 = 32 units
ANSWER:
P = 32 units; A = 56 units2
41.ALGEBRA The sides of a triangle have lengths x, x
+ 5, and 25. If the length of the longest side is 25,
what value of x makes the triangle a right triangle?
SOLUTION:
By the converse of the Pythagorean Theorem, if the
square of the longest side of a triangle is the sum of
squares of the other two sides then the triangle is a
right triangle.
Use the Quadratic Formula to find the roots of the
equation.
Since x is a length, it cannot be negative. Therefore,
x = 15.
ANSWER:
15
42.ALGEBRA The sides of a triangle have lengths 2x,
8, and 12. If the length of the longest side is 2x, what
values of x make the triangle acute?
SOLUTION:
By the triangle inequality theorem, the sum of the
lengths of any two sides should be greater than the
length of the third side.
So, the value of x should be between 2 and 10.
By the Pythagorean Inequality Theorem, if the
square of the longest side of a triangle is less than the
sum of squares of the other two sides then the
triangleisanacutetriangle.
Therefore, for the triangle to be acute,
ANSWER:
43.TELEVISION The screen aspect ratio, or the ratio
of the width to the length, of a high-definition
television is 16:9. The size of a television is given by
the diagonal distance across the screen. If an HDTV
is 41 inches wide, what is its screen size?
Refer to the photo on page 547.
SOLUTION:
Use the ratio to find the length of the television.
Let x be the length of the television. Then,
Solve the proportion for x.
The two adjacent sides and the diagonal of the
television form a right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Therefore, the screen size is about 47 inches.
ANSWER:
47 in.
44.PLAYGROUND According to the Handbook for
Public Playground Safety, the ratio of the vertical
distance to the horizontal distance covered by a slide
should not be more than about 4 to 7. If the horizontal
distance allotted in a slide design is 14 feet,
approximately how long should the slide be?
SOLUTION:
Use the ratio to find the vertical distance.
Let x be the vertical distance. Then,
Solve the proportion for x.
The vertical distance, the horizontal distance, and the
slide form a right triangle.
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
ofthehypotenuse.
Therefore, the slide will be about 16 ft long.
ANSWER:
about 16 ft
Find x.
45.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are x 4and8.
Solve for x.
ANSWER:
10
46.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
and the lengths of the legs are x 3and9.
Solve for x.
ANSWER:
15
47.
SOLUTION:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length
of the hypotenuse. The length of the hypotenuse is x
+ 1 and the lengths of the legs are x and .
Solve for x.
ANSWER:
48.MULTIPLE REPRESENTATIONS In this
problem, you will investigate special right triangles.
a. GEOMETRIC Draw three different isosceles
right triangles that have whole-number side lengths.
Label the triangles ABC, MNP, and XYZ with the
right angle located at vertex A, M, and X,
respectively. Label the leg lengths of each side and
find the length of the hypotenuse in simplest radical
form.
b. TABULAR Copy and complete the table below.
c. VERBAL Make a conjecture about the ratio of
the hypotenuse to a leg of an isosceles right triangle.
SOLUTION:
a.Let AB = AC. Use the Pythagorean Theorem to
find BC.
Sampleanswer:
 
Let MN = PM. Use the Pythagorean Theorem to
find NP.

Let ZX = XY. Use the Pythagorean Theorem to find
ZY.

b.Complete the table below with the side lengths
calculated for each triangle. Then, compute the given
ratios.
c. Summarize any observations made based on the
patternsinthetable.
Sample answer: The ratio of the hypotenuse to a leg
of an isosceles right triangle is .
ANSWER:
a.
b.
c. Sample answer: The ratio of the hypotenuse to a
leg of an isosceles right triangle is .
49.CHALLENGE Find the value of x in the figure.
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Find x****.

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 13 and the lengths of the legs are 5 and x. ANSWER: 12

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12.

ANSWER:

SOLUTION:

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12. ANSWER:

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 16 and the lengths of the legs are 4 and x. eSolutions Manual - Powered by Cognero Page 1 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

SOLUTION:

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 16 and the lengths of the legs are 4 and x. ANSWER:

  1. Use a Pythagorean triple to find x. Explain your reasoning. SOLUTION: 35 is the hypotenuse, so it is the greatest value in the Pythagorean Triple. Find the common factors of 35 and 21. The GCF of 35 and 21 is 7. Divide this out.

Check to see if 5 and 3 are part of a Pythagorean

triple with 5 as the largest value.

We have one Pythagorean triple 3- 4 - 5. The multiples

ANSWER:

  1. Use a Pythagorean triple to find x. Explain your reasoning. SOLUTION: 35 is the hypotenuse, so it is the greatest value in the Pythagorean Triple. Find the common factors of 35 and 21. The GCF of 35 and 21 is 7. Divide this out.

Check to see if 5 and 3 are part of a Pythagorean

triple with 5 as the largest value.

We have one Pythagorean triple 3- 4 - 5. The multiples of this triple also will be Pythagorean triple. So, x = 7 (4) = 28. Use the Pythagorean Theorem to check it. ANSWER: 28; Since and and 3- 4 - 5 is a Pythagorean triple, or 28.

  1. MULTIPLE CHOICE^ The mainsail of a boat is shown. What is the length , in feet , of? A 52. B (^) 65 C (^) 72. D (^) 75 eSolutions Manual - Powered by Cognero Page 2 8 - 2 The Pythagorean Theorem and Its Converse

Yes; right

  1. 16 , 18 , 26 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an obtuse triangle. ANSWER: Yes; obtuse
  2. 15 , 20 , 24 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an acute triangle.

ANSWER:

Yes; obtuse

  1. 15 , 20 , 24 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an acute triangle. ANSWER: Yes; acute Find x****.
  2. SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 12 and 16. eSolutions Manual - Powered by Cognero Page 4 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

Yes; acute Find x****.

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 12 and 16. ANSWER: 20

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 15 and the lengths of the legs are 9 and x. ANSWER: 12

ANSWER:

SOLUTION:

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 15 and the lengths of the legs are 9 and x. ANSWER: 12

SOLUTION: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 5 and the lengths of the legs are 2 and x. ANSWER: eSolutions Manual - Powered by Cognero Page 5 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

SOLUTION:

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is and the lengths of the legs are. ANSWER: CCSS PERSEVERANCE Use a Pythagorean Triple to find x****.

SOLUTION: Find the greatest common factors of 16 and 30. The GCF of 16 and 30 is 2. Divide this value out.

ANSWER:

CCSS PERSEVERANCE Use a Pythagorean Triple to find x****.

SOLUTION: Find the greatest common factors of 16 and 30. The GCF of 16 and 30 is 2. Divide this value out. Check to see if 8 and 15 are part of a Pythagorean triple. We have one Pythagorean triple 8- 15 - 17. The multiples of this triple also will be Pythagorean triple. So, x = 2(17) = 34. Use the Pythagorean Theorem to check it. ANSWER: 34

SOLUTION: Find the greatest common factor of 14 and 48. The GCF of 14 and 48 is 2. Divide this value out. Check to see if 7 and 24 are part of a Pythagorean eSolutions Manual - Powered by Cognero Page 7 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

SOLUTION:

Find the greatest common factor of 14 and 48. The GCF of 14 and 48 is 2. Divide this value out. Check to see if 7 and 24 are part of a Pythagorean triple. We have one Pythagorean triple 7- 24 - 25. The multiples of this triple also will be Pythagorean triple. So, x = 2(25) = 50. Use the Pythagorean Theorem to check it. ANSWER: 50

SOLUTION: 74 is the hypotenuse, so it is the greatest value in the Pythagorean triple. Find the greatest common factor of 74 and 24. The GCF of 24 and 74 is 2. Divide this value out. Check to see if 12 and 37 are part of a Pythagorean triple with 37 as the largest value.

ANSWER:

SOLUTION:

74 is the hypotenuse, so it is the greatest value in the Pythagorean triple. Find the greatest common factor of 74 and 24. The GCF of 24 and 74 is 2. Divide this value out. Check to see if 12 and 37 are part of a Pythagorean triple with 37 as the largest value. We have one Pythagorean triple 12- 35 - 37. The multiples of this triple also will be Pythagorean triple. So, x = 2(35) = 70. Use the Pythagorean Theorem to check it. ANSWER: 70

SOLUTION: 78 is the hypotenuse, so it is the greatest value in the Pythagorean triple. Find the greatest common factor of 78 and 72. 78 = 2 × 3 × 13 72 = 2 × 2 × 2 × 3 × 3 The GCF of 78 and 72 is 6. Divide this value out. 78 ÷ 6 = 13 72 ÷ 6 = 12 eSolutions Manual - Powered by Cognero Page 8 8 - 2 The Pythagorean Theorem and Its Converse

Therefore, the horizontal position of the support is about 3 ft. ANSWER: about 3 ft

  1. DRIVING^ The street that Khaliah usually uses to get to school is under construction. She has been taking the detour shown. If the construction starts at the point where Khaliah leaves her normal route and ends at the point where she re-enters her normal route , about how long is the stretch of road under construction? SOLUTION: Let x be the length of the road that is under construction. The road under construction and the detour form a right triangle. The length of the hypotenuse is x and the lengths of the legs are 0.8 and 1.8. Therefore, a stretch of about 2 miles is under construction. ANSWER: about 2 mi Determine whether each set of numbers can be the measures of the sides of a triangle. If so , classify the triangle as acute, obtuse, or righ t. Justify your answer.
  2. 7 , 15 , 21 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of construction. ANSWER: about 2 mi Determine whether each set of numbers can be the measures of the sides of a triangle. If so , classify the triangle as acute, obtuse, or righ t. Justify your answer.
  3. 7 , 15 , 21 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an obtuse triangle. ANSWER: Yes; obtuse
  4. 10 , 12 , 23 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Since the sum of the lengths of two sides is less than that of the third side, the set of numbers cannot be measures of a triangle. ANSWER: No; 23 > 10 + 12
  5. 4.5 , 20 , 20. SOLUTION: eSolutions Manual - Powered by Cognero Page 10 8 - 2 The Pythagorean Theorem and Its Converse

Yes; obtuse

  1. 10 , 12 , 23 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Since the sum of the lengths of two sides is less than that of the third side, the set of numbers cannot be measures of a triangle. ANSWER: No; 23 > 10 + 12
  2. 4.5 , 20 , 20. SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle. ANSWER: Yes; right
  3. 44 , 46 , 91 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side.

ANSWER:

Yes; right

  1. 44 , 46 , 91 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Since the sum of the lengths of two sides is less than that of the third side, the set of numbers cannot be measures of a triangle. ANSWER: No; 91 > 44 + 46
  2. 4.2 , 6.4 , 7. SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Therefore, the set of numbers can be measures of a triangle. Now, classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by Pythagorean Inequality Theorem, a triangle with the given measures will be an acute triangle. ANSWER: Yes; acute
  3. 4 , 12 , 14 SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. eSolutions Manual - Powered by Cognero Page 11 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

SOLUTION:

The segment of length 16 units is divided to two congruent segments. So, the length of each segment will be 8 units. Then we have a right triangle with the sides 15, 8, and x. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 15. ANSWER: 17

SOLUTION: We have a right triangle with the sides 14, 10, and x. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 14

ANSWER:

SOLUTION:

We have a right triangle with the sides 14, 10, and x. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The length of the hypotenuse is 14 and the lengths of the legs are 10 and x. ANSWER: COORDINATE GEOMETRY Determine whether is an acute, right, or obtuse triangle for the given vertices. Explain.

  1. X (– 3 , – 2) , Y (– 1 , 0) , Z (0 , – 1) SOLUTION: Use the distance formula to find the length of each eSolutions Manual - Powered by Cognero Page 13 8 - 2 The Pythagorean Theorem and Its Converse

ANSWER:

COORDINATE GEOMETRY Determine whether is an acute, right, or obtuse triangle for the given vertices. Explain.

  1. X (– 3 , – 2) , Y (– 1 , 0) , Z (0 , – 1) SOLUTION: Use the distance formula to find the length of each side of the triangle. Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Therefore, by the converse of Pythagorean Theorem, a triangle with the given measures will be a right triangle. ANSWER: right; , ,
  2. X (– 7 , – 3) , Y (– 2 , – 5) , Z (– 4 , – 1) SOLUTION: Use the distance formula to find the length of each si triangle. triangle. ANSWER: right; , ,
  3. X (– 7 , – 3) , Y (– 2 , – 5) , Z (– 4 , – 1) SOLUTION: Use the distance formula to find the length of each si triangle. Classify the triangle by comparing the square of the l the sum of the squares of the other two sides. Therefore, by the Pythagorean Inequality Theorem, a the given measures will be an acute triangle. ANSWER: acute; , , ;
  4. X (1 , 2) , Y (4 , 6) , Z (6 , 6) SOLUTION: Use the distance formula to find the length of each side of the triangle. eSolutions Manual - Powered by Cognero Page 14 8 - 2 The Pythagorean Theorem and Its Converse

a triangle with the given measures will be a right triangle. ANSWER: right; XY = 6, YZ = 10, XZ = 8; 6 2

  • 8 2 = 10 2
  1. JOGGING^ Brett jogs in the park three times a week. Usually , he takes a - mile path that cuts through the park. Today , the path is closed , so he is taking the orange route shown. How much farther will he jog on his alternate route than he would have if he had followed his normal path? SOLUTION: The normal jogging route and the detour form a right triangle. One leg of the right triangle is 0.45 mi. and let x be the other leg. The hypotenuse of the right triangle is. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. So, the total distance that he runs in the alternate route is 0.45 + 0. 6 = 1.05 mi. instead of his normal distance 0.75 mi. Therefore, he will be jogging an extra distance of 0. miles in his alternate route. ANSWER: 0.3 mi
  2. PROOF Write a paragraph proof of Theorem 8.5. SOLUTION: Theorem 8.5 states that if the sum of the squares of the lengths of the shortest sides of a triangle are equal to the square of the length of the longest side, then the triangle is a right triangle. Use the hypothesis to create a given statement, namely that c 2 = a 2 + b 2 for triangle ABC. You can accomplish this proof by the lengths of the shortest sides of a triangle are equal to the square of the length of the longest side, then the triangle is a right triangle. Use the hypothesis to create a given statement, namely that c 2 = a 2 + b 2 for triangle ABC. You can accomplish this proof by constructing another triangle (Triangle DEF) that is congruent to triangle ABC, using SSS triangle congruence theorem. Show that triangle DEF is a right triangle, using the Pythagorean theorem. therefore, any triangle congruent to a right triangle but also be a right triangle. Given: with sides of measure a , b , and c , where c 2 = a 2 + b 2 Prove: is a right triangle. Proof: Draw on line with measure equal to a. At D , draw line. Locate point F on m so that DF = b. Draw and call its measure x. Because is a right triangle, a 2 + b 2 = x 2 . But a 2 + b 2 = c 2 , so x 2 = c 2 or x = c. Thus, by SSS. This means. Therefore, C must be a right angle, making a right triangle. ANSWER: Given: with sides of measure a , b , and c , where c 2 = a 2 + b 2 Prove: is a right triangle. Proof: Draw on line with measure equal to a. At D , draw line. Locate point F on m so that DF = b. Draw and call its measure x. Because is a right triangle, a 2 + b 2 = x 2 . But a 2 + b 2 = c 2 , so x 2 = c 2 or x = c. Thus, by SSS. This means. eSolutions Manual - Powered by Cognero Page 16 8 - 2 The Pythagorean Theorem and Its Converse

Proof: Draw on line with measure equal to a. At D , draw line. Locate point F on m so that DF = b. Draw and call its measure x. Because is a right triangle, a 2

  • b 2 = x 2

. But a 2

  • b 2 = c 2 , so x 2 = c 2 or x = c. Thus, by SSS. This means. Therefore, C must be a right angle, making a right triangle. PROOF Write a two-column proof for each theorem.
  1. Theorem 8. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two triangles, a right angle, relationship between sides. Use the properties that you have learned about right angles, acute angles, Pythagorean Theorem,angle relationships and equivalent expressions in algebra to walk through the proof. Given: In , c 2 < a 2
  • b 2 where c is the length of the longest side. In , R is a right angle. Prove: is an acute triangle. Proof: Statements (Reasons)
  1. In , c 2 < a 2
  • b 2 where c is the length of the longest side. In , R is a right angle. (Given)
  1. a 2
  • b 2 = x 2 (Pythagorean Theorem)
  1. c 2 < x 2 (Substitution Property)
  2. c < x (A property of square roots)
  3. m R = (Definition of a right angle)
  4. m C < m R (Converse of the Hinge Theorem)
  5. m C < (Substitution Property)
  6. C is an acute angle. (Definition of an acute angle)
  7. is an acute triangle. (Definition of an acute triangle) ANSWER: Given: In , c 2 < a 2
  • b 2 where c is the length of the longest side. In , R is a right angle.
  1. C is an acute angle. (Definition of an acute angle)
  2. is an acute triangle. (Definition of an acute triangle) ANSWER: Given: In , c 2 < a 2
  • b 2 where c is the length of the longest side. In , R is a right angle. Prove: is an acute triangle. Proof: Statements (Reasons)
  1. In , c 2 < a 2
  • b 2 where c is the length of the longest side. In , R is a right angle. (Given)
  1. a 2
  • b 2 = x 2 (Pythagorean Theorem)
  1. c 2 < x 2 (Substitution Property)
  2. c < x (A property of square roots)
  3. m R = (Definition of a right angle)
  4. m C < m R (Converse of the Hinge Theorem)
  5. m C < (Substitution Property)
  6. C is an acute angle. (Definition of an acute angle)
  7. is an acute triangle. (Definition of an acute triangle)
  8. Theorem 8. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two triangles, relationship between angles.Use the properties that you have learned about triangles, angles and equivalent expressions in algebra to walk through the proof. Given: In , c 2

a 2

  • b 2 , where c is the length of the longest side. Prove: is an obtuse triangle. Statements (Reasons)
  1. In , c 2

a 2

  • b 2 , where c is the length of the eSolutions Manual - Powered by Cognero Page 17 8 - 2 The Pythagorean Theorem and Its Converse

the perimeter is 12 + 16 + 20 = 48 units. ANSWER: P = 48 units; A = 96 units 2

SOLUTION: The area of a triangle is given by the formula where b is the base and h is the height of the triangle. The altitude to the base of an isosceles triangle bisects the base. So, we have two right triangles with one of the legs equal to 5 units and the hypotenuse is 13 units each. Use the Pythagorean Theorem to find the length of the common leg of the triangles. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The altitude is 12 units. Therefore, The perimeter is the sum of the lengths of the three sides. Therefore, the perimeter is 13 + 13 + 10 = 36 units. ANSWER: P = 36 units; A = 60 units 2 sides. Therefore, the perimeter is 13 + 13 + 10 = 36 units. ANSWER: P = 36 units; A = 60 units 2

SOLUTION: The given figure can be divided as a right triangle and a rectangle as shown. The total are of the figure is the sum of the areas of the right triangle and the rectangle. The area of a triangle is given by the formula where b is the base and h is the height of the triangle. Since the triangle is a right triangle the base and the height are the legs of the triangle. So, The area of a rectangle of length l and width w is given by the formula A = l × w. So, Therefore, the total area is 24 + 32 = 56 sq. units. The perimeter is the sum of the lengths of the four boundaries. Use the Pythagorean Theorem to find the length of the hypotenuse of the triangle. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The hypotenuse is 10 units. Therefore, the perimeter is 4 + 8 + 10 + 10 = 32 units ANSWER: 2 eSolutions Manual - Powered by Cognero Page 19 8 - 2 The Pythagorean Theorem and Its Converse

The hypotenuse is 10 units. Therefore, the perimeter is 4 + 8 + 10 + 10 = 32 units ANSWER: P = 32 units; A = 56 units 2

  1. ALGEBRA^ The sides of a triangle have lengths x, x
    • 5 , and 25. If the length of the longest side is 25 , what value of x makes the triangle a right triangle? SOLUTION: By the converse of the Pythagorean Theorem, if the square of the longest side of a triangle is the sum of squares of the other two sides then the triangle is a right triangle. Use the Quadratic Formula to find the roots of the equation. Since x is a length, it cannot be negative. Therefore, x = 15. ANSWER: 15
  2. ALGEBRA^ The sides of a triangle have lengths 2 x, 8 , and 12. If the length of the longest side is 2 x, what values of x make the triangle acute? SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. Since x is a length, it cannot be negative. Therefore, x = 15. ANSWER: 15
  3. ALGEBRA^ The sides of a triangle have lengths 2 x, 8 , and 12. If the length of the longest side is 2 x, what values of x make the triangle acute? SOLUTION: By the triangle inequality theorem, the sum of the lengths of any two sides should be greater than the length of the third side. So, the value of x should be between 2 and 10. By the Pythagorean Inequality Theorem, if the square of the longest side of a triangle is less than the sum of squares of the other two sides then the triangle is an acute triangle. Therefore, for the triangle to be acute, ANSWER:
  4. TELEVISION The screen aspect ratio , or the ratio of the width to the length , of a high-definition television is 16:9. The size of a television is given by the diagonal distance across the screen. If an HDTV is 41 inches wide , what is its screen size?

Refer to the photo on page 547.

SOLUTION:

Use the ratio to find the length of the television. Let x be the length of the television. Then, eSolutions Manual - Powered by Cognero Page 20 8 - 2 The Pythagorean Theorem and Its Converse