# If c is the measure of the hypotenuse of a right triangle, how do you find each missing measure given a=x, b=x+41, c=85?

Mar 7, 2017

36 & 77

#### Explanation:

We know, In a right triangle, $h y p o t e \nu s {e}^{2} = b a s {e}^{2} + h e i g h {t}^{2}$

Here, ${c}^{2} = {a}^{2} + {b}^{2}$

$\Rightarrow {85}^{2} = {x}^{2} + {\left(x + 41\right)}^{2}$

$\Rightarrow 7225 = {x}^{2} + {x}^{2} + 82 x + 1681$

$\Rightarrow 2 {x}^{2} + 82 x + 1681 - 7225 = 0$

$\Rightarrow$ 2x^2 + 82x - 5544=0

$\Rightarrow 2 \left({x}^{2} + 41 x - 2772\right) = 0$

$\Rightarrow {x}^{2} + 41 x - 2772 = 0$

$\Rightarrow {x}^{2} + 77 x - 36 x - 2772 = 0$

$\Rightarrow x \left(x + 77\right) - 36 \left(x + 77\right) = 0$

$\Rightarrow \left(x - 36\right) \left(x + 77\right) = 0$

rArr x = 36 & -77# $\left[x \ne - 77\right]$

Hence a = 36 & b = 36+41 = 77