Question

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

Short answer

The value is, $x=8,y=8\sqrt{2},z=8\sqrt{2}$.

Step-by-step solution

Step 1. Given information.

The given figure is,

Step 2. Use geometric mean concept for x.

$x$ is the geometric mean of $8$ and $8$.

So,

$\begin{array}{l}x=\sqrt{8\cdot 8}\\ =8\end{array}$

Step 3. Now, Use Pythagoras Theorem in right angled triangles to find y and z :

$\begin{array}{c}{z}^2=8^2+8^2\\ z^2=2\cdot 8^2\\ z=\sqrt{2\cdot 8^2}[\text{takesquarerooteachside]}\\ z=8\sqrt{2}\end{array}$

$\begin{array}{c}{y}^2=8^2+8^2\\ y^2=2\cdot 8^2\\ y=\sqrt{2\cdot 8^2}[\text{takesquarerooteachside]}\\ y=8\sqrt{2}\end{array}$

Therefore, the value is $x=8,y=8\sqrt{2},z=8\sqrt{2}$.