Question

If A(5,2), B(2,-2) and C(-2,r) are the vertices of a right angled triangle with

Answer 1

A(5, 2)    B (2, -2)    C(-2,  r)

We don't know which angle is 90° in the ΔABC.

Slope of AB: (-2-2)/(2-5) =4/3           Slope of BC : (r+2)/(-2-2) = -(r+2)/4
Slope of CA:  (2-r)/(5--2) = (2-r)/7
AB² = (5-2)²+(2--2)² = 25 ,           BC² = (2--2)²+(-2-r)²=r²+4r+20
CA² = (5--2)²+(2-r)²= r² - 4r +53
case 1) Let AB ⊥ BC ,  ∠B = 90°
            Product of slopes:  4/3 * -(r+2)/4 = -1    => r = 1
             check:  AB² + BC² = 25+r² + 4r + 20 = r²-4r+53  => r = 1
case 2) Let   AB ⊥ CA     ∠A = 90°
            Product of slopes= 4/3 *(2-r)/7 = -1   => r = 29/4
            AB²+ CA² = 25+ r² -4r+53 = r²+4r+20  => r = 58/8 = 29/4

case 3) AC ⊥  BC  ,     
                 slopes:    -(r+2)/4 * (2-r)/7 = -1   =>  r²  is negative,
                So it is not possible.

So r can have two values possible.