Question

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

Explanation:

Step1: Identify geometric theorem

This is a right triangle altitude scenario, so we use the geometric mean (altitude-on-hypotenuse) theorem: the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments it divides the hypotenuse into. Additionally, each leg is the geometric mean of the hypotenuse and the adjacent segment. Here, we use the leg relationship:
$$(x+19)^2 = x \times 76$$

Step2: Expand left-hand side

$$x^2 + 38x + 361 = 76x$$

Step3: Rearrange to quadratic equation

$$x^2 + 38x + 361 - 76x = 0$$
$$x^2 - 38x + 361 = 0$$

Step4: Factor the quadratic

Notice this is a perfect square trinomial:
$$(x - 19)^2 = 0$$

Step5: Solve for x

$$x - 19 = 0$$
$$x = 19$$

Answer:

19