Chapter 8.1, Problem 36WE

To determine

Find x , y , and z .

Answer to Problem 36WE

  $x=18$

  $y=12\sqrt{2}$

  $z=4\sqrt{2}$

Explanation of Solution

Given:

  

Calculation:

The small right triangle with hypotenuse 6 is similar to the large right triangle with hypotenuse x .

So,

  $\begin{array}{l}\frac{6}{x}=\frac{2}{6}\\ \Rightarrow 6^2=2x\mathrm{...................}[\text{multiply each side by 6x]}\\ \Rightarrow x=\frac{36}{2}\mathrm{.....................}[\text{divide each side by 2]}\\ \Rightarrow x=18\end{array}$

Find y by Pythagoras Theorem:

  $\begin{array}{l}{y}^2+6^2={18}^2\\ \Rightarrow y^2=324-\mathrm{36...........}[\text{subtract 36 from each side]}\\ \Rightarrow y=\sqrt{288}\mathrm{.................}[\text{take square root each side]}\\ \Rightarrow y=\sqrt{2^2\cdot 6^2\cdot 2}\\ \Rightarrow y=12\sqrt{2}\end{array}$

z is the geometric mean of 2 and 18-2 :

  $\begin{array}{l}z=\sqrt{2\cdot 16}\\ \Rightarrow z=4\sqrt{2}\end{array}$

So,

  $x=\frac{27}{5}$

  $y=\frac{48}{5}$

  $z=\frac{36}{5}$