if A(5,2), B(2,-2) and C (-2,t) are the vertices of right angled triangle with $∠ B = 90^{0},$ then find the value of t.

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Solution

Using Pythagoras theorem in right triangle ABC, we obtain
${A C}^{2} = {A B}^{2} + {B C}^{2}$
$\Rightarrow {(5 + 2)}^{2} + {(2 - t)}^{2} = {{(5 - 2)}^{2} + {(2 + 2)}^{2}} + {{(2 + 2)}^{2} + {(- 2 - t)}^{2}}$
$\Rightarrow 49 + (4 - 4 t + t^{2}) = (9 + 16) + (16 + 4 + 4 t + t^{2})$
$\Rightarrow t^{2} - 4 t + 53 = t^{2} + 4 t \times 45$
$\Rightarrow - 8 t = - 8$
$\Rightarrow t = 1$

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