Question

5 fill in the blank 2 points solve for the variable. leave your answer as a simplified radical. 3x 50 2x

Explanation:

Step1: Identify similar triangles

The two smaller right triangles are similar to the large right triangle, so we can use the geometric mean (altitude-on-hypotenuse) theorem. For a right triangle, the length of the altitude to the hypotenuse is equal to the geometric mean of the lengths of the two segments it creates on the hypotenuse. Additionally, each leg is the geometric mean of the hypotenuse and the adjacent segment.
For leg length $50$, it is the geometric mean of $3x$ and the total hypotenuse length $3x + 2x = 5x$.

Step2: Set up the proportion

$$50^2 = 3x \times 5x$$

Step3: Simplify and solve for $x$

First calculate $50^2$ and simplify the right-hand side:
$$2500 = 15x^2$$
Rearrange to solve for $x^2$:
$$x^2 = \frac{2500}{15} = \frac{500}{3}$$
Take the square root of both sides (since length is positive):
$$x = \sqrt{\frac{500}{3}} = \frac{\sqrt{500}}{\sqrt{3}} = \frac{10\sqrt{5}}{\sqrt{3}}$$
Rationalize the denominator:
$$x = \frac{10\sqrt{15}}{3}$$

Answer:

$\boldsymbol{\frac{10\sqrt{15}}{3}}$