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Finding the Third Vertex of a Triangle Given the Orthocenter and Centroid, Exams of Mathematics

A solution to finding the third vertex of a triangle when the orthocenter and centroid are given. The triangle has two vertices at (-2,3) and (5,-1), and the orthocenter is at the origin. The centroid lies on the line x+y=7. By using the property that the line passing through the third vertex and the orthocenter must be perpendicular to the line through the other two vertices, the document deduces that the third vertex does not match any of the given options (a, b, or c).

Typology: Exams

2017/2018

Uploaded on 01/07/2018

Sujeendra
Sujeendra 🇮🇳

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1. If two vertices of a triangle are (-2,3) and (5,-1). Orthocentre of the triangle lies at the
origin and centroid on the line x+y=7, then the third vertex lies at
A. (7,4)
B. (8,14)
C. (12,21)
D. None of these
Ans: D
Solution: The line passing through the third vertex and orthocentre must be perpendicular to
line through (-2,3) and (5,-1) .Therefore the product of their slope is -1.
Given the two vertices B(-2,3) and C(5,-1); let H(0,0) be the orthocentre ;A(h,k)
Be the third vertex. Then slope of the line through A and H is k/h, while the line
through B & C has the slope (-1-3)/(5+2)=-4/7. By the property of the
orthocentre ,these two lines must be perpendicular ,

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1. If two vertices of a triangle are (-2,3) and (5,-1). Orthocentre of the triangle lies at the

origin and centroid on the line x+y=7, then the third vertex lies at

A. (7,4)

B. (8,14)

C. (12,21)

D. None of these

Ans: D Solution: The line passing through the third vertex and orthocentre must be perpendicular to line through (-2,3) and (5,-1) .Therefore the product of their slope is -1. Given the two vertices B(-2,3) and C(5,-1); let H(0,0) be the orthocentre ;A(h,k) Be the third vertex. Then slope of the line through A and H is k/h, while the line through B & C has the slope (-1-3)/(5+2)=-4/7. By the property of the orthocentre ,these two lines must be perpendicular ,