Question

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

Short answer

Value of,

$\begin{array}{c}x=\sqrt{2}\\ y=2\\ z=\sqrt{2}\end{array}$

Step-by-step solution

Step 1. Given information.

The given figure is,

Step 2. Concept used.

The small right triangle with hypotenuse $2$ is similar to the large right triangle with hypotenuse $\sqrt{2}+x$.

So,

$\begin{array}{c}\frac{2}{\sqrt{2}+x}=\frac{\sqrt{2}}{2}\\ 2^2=\sqrt{2}\left(\sqrt{2}+x\right)[\text{multiplyeachsideby2}\left(\sqrt{2}+x\right)\text{]}\\ 4=2+\sqrt{2}x\\ \sqrt{2}x=2[\text{subtract2fromeachside]}\\ x=\frac{2}{\sqrt{2}}[\text{divideeachsideby}\sqrt{2}\text{]}\\ x=\sqrt{2}\end{array}$

Step 3. Use geometric mean concept.

$z$ is the geometric mean of $\sqrt{2}$ and $x$.

$\begin{array}{c}z=\sqrt{\sqrt{2}\cdot x}\\ z=\sqrt{\sqrt{2}\cdot \sqrt{2}}\\ z=\sqrt{2}\end{array}$

The small right triangle with hypotenuse $y$ is similar to the large right triangle with hypotenuse $\sqrt{2}+x$ .

So,

$\begin{array}{c}\frac{y}{\sqrt{2}+x}=\frac{x}{y}\\ y^2=x\left(\sqrt{2}+x\right)\mathrm{...................}[\text{multiplyeachsidebyy}\left(\sqrt{2}+x\right)\text{]}\\ y^2=\sqrt{2}\left(\sqrt{2}+\sqrt{2}\right)\\ y^2=\sqrt{2}\left(2\sqrt{2}\right)\\ y^2=4\\ y=\mathrm{2................................}[\text{takesquarerooteachside]}\end{array}$

Therefore, the value of,

$\begin{array}{c}x=\sqrt{2}\\ y=2\\ z=\sqrt{2}\end{array}$