Question

if A(9,-9). (1. -3) are the end points of hypotenuse of a right angled isosceles triangle, then the third vertex​

Answer 1

Given : A(9,-9). C (1. -3) are the end points of hypotenuse of a right angled isosceles triangle

To Find :  the third vertex​

Solution:

A(9,-9). C (1. -3) are the end points of hypotenuse of a right angled isosceles triangle

Hence median at hypotenuse will be perpendicular

Let say  B = (x , y)

AB ⊥ BC

Slope of AB * Slope of BC = - 1

=>  ( y + 9)/(x - 9)   * ( y + 3)/(x - 1)  = -1

=> y² + 12y + 27  = -(x²  - 10x + 9)

=>  x² + y² - 10x  + 12y  + 36 = 0

AB = BC  given

=> (x - 9)² + (y +9)² =  (x -1)² + (y + 3)²

=> x² - 18x + 81 + y²+ 18y + 81 = x² -2x + 1 + y² + 6y + 9

=> 16x - 12y = 152

=> 4x - 3y = 38

4x - 3y = 38

x² + y² - 10x  + 12y  + 36 = 0

on solving   u get  x = 2 and 8

third vertex​   ( 2 , - 10) or (8 , -2 )

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