Question

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

Short answer

The value is $x=10,y=2\sqrt{29},z=5\sqrt{29}$.

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 $4$ and $25$.

So,

$\begin{array}{c}x=\sqrt{4\cdot 25}\\ =\sqrt{2^2\cdot 5^2}\\ =10\end{array}$

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

Substitute the values and calculate.

$\begin{array}{c}{z}^2={25}^2+{10}^2\\ z^2=625+100\\ z^2=725\\ z=\sqrt{25\cdot 29}\mathrm{..............}[\text{takesquarerooteachside]}\\ z=5\sqrt{29}\end{array}$

And,

$\begin{array}{c}{y}^2=4^2+{10}^2\\ y^2=16+100\\ y^2=116\\ y=\sqrt{4\cdot 29}\mathrm{..............}[\text{takesquarerooteachside]}\\ y=2\sqrt{29}\end{array}$

Therefore, the value is $x=10,y=2\sqrt{29},z=5\sqrt{29}$.