Answer
$x = 10$
Work Step by Step
With similar triangles, corresponding sides are proportional.
Let's set up a proportion using the two sets of corresponding sides that we are given:
$\frac{10}{x + 10} = \frac{x + 2}{x + 14}$
Cross multiply to get rid of the fractions:
$(x + 10)(x + 2) = 10(x + 14)$
Multiply the terms out:
$x^2 + 10x + 2x + 20 = 10x + 140$
Combine like terms:
$x^2 + 12x + 20 = 10x + 140$
Subtract $10x$ from each side of the equation to move variables to the left side of the equation:
$x^2 + 2x + 20 = 140$
Subtract $140$ from each side of the equation to move constants to the left side of the equation:
$x^2 + 2x - 120 = 0$
Factor the quadratic equation:
$(x + 12)(x - 10) = 0$
Set each factor equal to $0$ to solve for $x$:
$x + 12 = 0$
Subtract $12$ from each side of the equation to solve:
$x = -12$
We discard this solution because $x$ cannot be negative.
Let's look at the other factor:
$x - 10 = 0$
Add $10$ to each side of the equation:
$x = 10$