# The hypotenuse of an isosceles right triangle has endpoints (4,3) and (9,8). What is the length of one of the legs of the triangles?

Mar 12, 2018

$5$.

#### Explanation:

Suppose that in the isosceles right- $\Delta A B C , \angle B = {90}^{\circ}$.

So $A C$ is the hypotenuse, and we take, A(4,3) & C(9,8).

Clearly, we have, $A B = B C \ldots \ldots \ldots \ldots \ldots \ldots \left(\ast\right)$.

Applying Pythagoras Theorem, we have,

$A {B}^{2} + B {C}^{2} = A {C}^{2} = {\left(4 - 9\right)}^{2} + {\left(3 - 8\right)}^{2}$.

$\therefore B {C}^{2} + B {C}^{2} = 25 + 25 = 50$.

$\therefore 2 B {C}^{2} = 50$.

$\therefore B C = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$.

$\Rightarrow A B = B C = 5$.