Chapter 9, Problem 12CUR

To determine

To Find: The length of the shorter leg.

Answer to Problem 12CUR

The length of the shorter side is 12.

Explanation of Solution

Given:

The length of the segments are 8 and 10.

Calculation:

Consider the triangle is $ABC$ . Then, the relation is,

  $\begin{array}{l}{\left(BC\right)}^2={\left(AB\right)}^2+{\left(AC\right)}^2\\ \left(8+10\right)=x^2+y^2\\ x^2+y^2={18}^2\end{array}$

Consider the point on the hypotenuse is $D$ then $\Delta ABD$ relation is,

  $x^2=a^2+8^2$

Consider the point on the hypotenuse is $D$ then $\Delta ADC$ relation is,

  $y^2=a^2+{\left(10\right)}^2$

From above equations,

  $\begin{array}{l}{x}^2+y^2=2a^2+\left({10}^2\right)+8^2\\ {18}^2=2a^2+\left({10}^2\right)+8^2\\ 2a^2=160\\ a^2=80\end{array}$

Then, the length of the side $x$ is,

  $\begin{array}{l}{x}^2=80+8^2\\ x=12\end{array}$

Then, the length of the side $y$ is,

  $\begin{array}{l}{y}^2=80+100\\ y=13.4164\end{array}$

Thus, the length of the shorter side is 12.