find the value of x using the figure to the right.\nx = (simplify your answer.)

Explanation:

Step1: Use the geometric - mean theorem

In a right - triangle, if an altitude is drawn to the hypotenuse, then the length of each leg of the right - triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Let the hypotenuse be (x + 14). The two segments of the hypotenuse are (x) and (14), and the altitude is (21). We know that (21^{2}=14x) (by the geometric - mean theorem: altitude squared = product of the two segments of the hypotenuse).

Step2: Solve the equation for (x)

First, expand the left - hand side of the equation: (441 = 14x). Then, divide both sides of the equation by (14): (x=\frac{441}{14}=\frac{63}{2}=31.5).

Answer:

(31.5)