Question

Find the values of $x,y,$and $z$.

Short answer

Value of,

$\begin{array}{c}x=9\\ y=15\\ x=20\end{array}$

Step-by-step solution

Step 1. Given information.

The given figure is,

Step 2. Use geometric mean concept.

$12$ is the geometric mean of $x$ and $x+7$.

$\begin{array}{c}12=\sqrt{x(x+7)}\\ {12}^2=x^2+7x[\text{squareeachside]}\\ x^2+7x-144=0[\text{subtract144fromeachside]}\\ x=\frac{-7\pm \sqrt{7^2-4\left(1\right)(-144)}}{2\left(1\right)}\\ x=\frac{-7\pm \sqrt{625}}{2}\\ x=\frac{-7\pm 25}{2}\\ x=\frac{-7+25}{2}[\text{sincelengthispositive]}\\ \Rightarrow x=9\end{array}$

Step 3. Concept used.

The small right triangle with hypotenuse $y$ is similar to the large right triangle with hypotenuse $2x+7$.

So,

$\begin{array}{c}\frac{y}{2x+7}=\frac{x}{y}\\ y^2=x\left(2x+7\right)[\text{multiplyeachsidebyy}\left(2x+7\right)\text{]}\\ y^2=9\left(2\left(9\right)+7\right)\\ y=\sqrt{9\cdot 25}[\text{takesquarerooteachside]}\\ y=15\end{array}$

The small right triangle with hypotenuse $z$ is similar to the large right triangle with hypotenuse $2x+7=25$ .

So,

$\begin{array}{c}\frac{z}{25}=\frac{x+7}{z}\\ z^2=25\left(9+7\right)[\text{multiplyeachsideby25z]}\\ z^2=25\left(16\right)\\ z=20[\text{takesquarerooteachside]}\end{array}$

Therefore, the value of,

$\begin{array}{c}x=9\\ y=15\\ z=20\end{array}$