Question

solve for $x$. leave your answer in simplest radical form.

Explanation:

Step1: Define shared segment as $y$

Let the length of the shared segment (the common side of the two right triangles) be $y$.

Step2: Use Pythagoras on top triangle

For the upper right triangle: $y^2 + 7^2 = 9^2$
$y^2 = 9^2 - 7^2 = 81 - 49 = 32$

Step3: Use Pythagoras on bottom triangle

For the lower right triangle: $x^2 + 2^2 = y^2$
Substitute $y^2=32$: $x^2 = 32 - 4 = 28$

Step4: Simplify radical for $x$

$x = \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$

Answer:

$2\sqrt{7}$