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Two vertices of a triangle are (4,-3) and (-2,5), if the orthocentre of the triangle is (1,2). Find the coordinates of the third vertex.

Two vertices of a triangle are (4,-3) and (-2,5), if the orthocentre of the triangle is (1,2). Find the coordinates of the third vertex.

Grade:12

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
Suppose the points are A (4, -3) and B (-2, 5), with the orthocenter O (1, 2). We want to find the 3rd vertex of the triangle, C (x, y).
The slope of the altitude coming out of A is= -5/3
Side BC must have slope 3/5, since it's perpendicular to this altitude, and perpendicular lines have slopes that are negative reciprocals.
So the equation for the line through B and C is y = (3/5)x + b, and when you substitute the known point (-2, 5) into this, we find b =31/5. So the equation of the line is 5y = 3x +31.
Now do the same thing for the altitude coming out of B. The slope is -1, so the slope of side AC must be 1. y =x + b, and plugging in (4, -3), we find b = -7.
So y=x-7
We now have two lines that include BC and AC, respectively, and we can find point C by solving for the intersection of these two lines.
5y = 3x+31
y = x-7
By solving the above 2 equations we get C = (33,26).

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