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
ofthehypotenuse.
The length of the hypotenuse is x and the lengths of
thelegsare8and12.
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.
TheGCFof35and21is7.Dividethisout.
Check to see if 5 and 3 are part of a Pythagorean
triplewith5asthelargestvalue.
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
legsare45and60.
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
andthelengthsofthelegsare12and16.
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
andthelengthsofthelegsare .
ANSWER:
CCSSPERSEVERANCEUseaPythagorean
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:
Findthegreatestcommonfactorof14and48.
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.
Findthegreatestcommonfactorof74and24.
The GCF of 24 and 74 is 2. Divide this value out.
Check to see if 12 and 37 are part of a Pythagorean
triplewith37asthelargestvalue.
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
andthelengthsofthelegsare
.
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
thelegsare0.8and1.8.
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
andthelengthsofthelegsare9and12.
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
andthelengthsofthelegsare8and15.
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
fortriangleABC.Youcanaccomplishthisproofby
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
butalsobearighttriangle.
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 < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofan
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 < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofanacute
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 = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
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 = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
CCSS SENSE-MAKINGFindtheperimeter
and area of each figure.
38.
SOLUTION:
The area of a triangle is given by the formula
whereb 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
ofthehypotenuse.
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
whereb 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
ofthehypotenuse.
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
whereb 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
ofthehypotenuse.
Thehypotenuseis10units.
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
triangleisanacutetriangle.
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
ofthehypotenuse.
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
ofthehypotenuse.
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 – 4and8.
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 – 3and9.
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.
Sampleanswer:
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
patternsinthetable.
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
ofthehypotenuse.
The length of the hypotenuse is x and the lengths of
thelegsare8and12.
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.
TheGCFof35and21is7.Dividethisout.
Check to see if 5 and 3 are part of a Pythagorean
triplewith5asthelargestvalue.
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
legsare45and60.
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
andthelengthsofthelegsare12and16.
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
andthelengthsofthelegsare .
ANSWER:
CCSSPERSEVERANCEUseaPythagorean
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:
Findthegreatestcommonfactorof14and48.
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.
Findthegreatestcommonfactorof74and24.
The GCF of 24 and 74 is 2. Divide this value out.
Check to see if 12 and 37 are part of a Pythagorean
triplewith37asthelargestvalue.
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
andthelengthsofthelegsare
.
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
thelegsare0.8and1.8.
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
andthelengthsofthelegsare9and12.
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
andthelengthsofthelegsare8and15.
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
fortriangleABC.Youcanaccomplishthisproofby
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
butalsobearighttriangle.
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 < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofan
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 < (SubstitutionProperty)
8. C is an acute angle. (Definition of an acute
angle)
9. isanacutetriangle.(Definitionofanacute
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 = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
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 = (Definitionofarightangle)
6. m C > m R (Converse of the Hinge Theorem)
7. m C > (SubstitutionPropertyofEquality)
8. C is an obtuse angle. (Definition of an obtuse
angle)
9. isanobtusetriangle.(Definitionofan
obtuse triangle)
CCSS SENSE-MAKINGFindtheperimeter
and area of each figure.
38.
SOLUTION:
The area of a triangle is given by the formula
whereb 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
ofthehypotenuse.
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
whereb 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
ofthehypotenuse.
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
whereb 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
ofthehypotenuse.
Thehypotenuseis10units.
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
triangleisanacutetriangle.
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
ofthehypotenuse.
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
ofthehypotenuse.
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 – 4and8.
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 – 3and9.
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.
Sampleanswer:
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
patternsinthetable.
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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