find the value of $x$. leave your answer in simplest radical form.

Explanation:

Step1: Find top rectangle side length

Use Pythagorean theorem on top right triangle: Let the horizontal side of the rectangle be $s$. $\begin{align} s &= \sqrt{4^2 + 2^2} \ &= \sqrt{16 + 4} \ &= \sqrt{20} = 2\sqrt{5} \end{align}$

Step2: Solve for $x$ using Pythagoras

Use Pythagorean theorem on the rectangle with diagonal 10: $\begin{align} x &= \sqrt{10^2 - s^2} \ &= \sqrt{100 - (2\sqrt{5})^2} \ &= \sqrt{100 - 20} \ &= \sqrt{80} \end{align}$

Step3: Simplify the radical

$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

Answer:

$4\sqrt{5}$