Chapter 8.1, Problem 38WE

To determine

Find x , y , and z .

Answer to Problem 38WE

  $x=9$

  $y=15$

  $z=20$

Explanation of Solution

Given:

  

Calculation:

12 is the geometric mean of x and x + 7:

  $\begin{array}{l}12=\sqrt{x(x+7)}\\ \Rightarrow {12}^2=x^2+7x\mathrm{............}[\text{square each side]}\\ \Rightarrow x^2+7x-144=\mathrm{0......}[\text{subtract 144 from each side]}\\ \Rightarrow x=\frac{-7\pm \sqrt{7^2-4(1)(-144)}}{2(1)}\\ \Rightarrow x=\frac{-7\pm \sqrt{625}}{2}\\ \Rightarrow x=\frac{-7\pm 25}{2}\\ \Rightarrow x=\frac{-7+25}{2}\mathrm{...........}[\text{since length is positive]}\\ \Rightarrow x=9\end{array}$

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

So,

  $\begin{array}{l}\frac{y}{2x+7}=\frac{x}{y}\\ \Rightarrow y^2=x\left(2x+7\right)\mathrm{...................}[\text{multiply each side by y}\left(2x+7\right)\text{]}\\ \Rightarrow y^2=9\left(2(9)+7\right)\\ \Rightarrow y=\sqrt{9\cdot 25}\mathrm{..........................}[\text{take square root each side]}\\ \Rightarrow y=15\end{array}$

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

So,

  $\begin{array}{l}\frac{z}{25}=\frac{x+7}{z}\\ \Rightarrow z^2=25\left(9+7\right)\mathrm{...................}[\text{multiply each side by 25z]}\\ \Rightarrow z^2=25\left(16\right)\\ \Rightarrow z=\mathrm{20................................}[\text{take square root each side]}\end{array}$

So,

  $x=9$

  $y=15$

  $z=20$