Answer
$x = 4$
Work Step by Step
We have a right triangle with the altitutde dividing the triangle into two smaller triangles; therefore, these three triangles are all similar to one another.
If these triangles are similar, then their corresponding sides are proportional.
We are given either values or expressions for the lengths of one of the legs and the hypotenuse of the big triangle and the lengths of one of the legs and the hypotenuse of the smaller triangle. Let's set up a proportion to find the value of $x$:
$\frac{x}{x + 2} = \frac{x + 2}{x+ 5}$
Cross multiply to get rid of the fractions:
$x(x + 5) = (x + 2)(x + 2)$
Distribute to simplify:
$x^2 + 5x = x^2 + 2x + 2x + 4$
Move all terms to the left side of the equation:
$x^2 + 5x - x^2 - 2x - 2x - 4 = 0$
Combine like terms:
$x - 4 = 0$
Add $4$ to each side of the equation to solve for $x$:
$x = 4$
