Question

find the values of the variables.
x=
y=
z=
options: $3\sqrt{6}$, $12$, $12\sqrt{5}$, $6\sqrt{5}$, $6\sqrt{3}$, $2\sqrt{13}$, $6$, $3\sqrt{13}$, $13$

Explanation:

Step1: Find x via geometric mean

In a right triangle, the length of a leg is the geometric mean of the hypotenuse segment adjacent to the leg and the full hypotenuse. The hypotenuse is $x + 9$, so:
$$x = \frac{6^2}{9} = \frac{36}{9} = 4$$
(matches the given x value)

Step2: Calculate full hypotenuse length

Sum x and the given segment:
$$x + 9 = 4 + 9 = 13$$

Step3: Find y via Pythagorean theorem

Use right triangle with legs y, x, altitude 6:
$$y = \sqrt{x^2 + 6^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$$

Step4: Find z via Pythagorean theorem

Use right triangle with legs z, 9, altitude 6:
$$z = \sqrt{9^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117} = 3\sqrt{13}$$

Answer:

$x=4$, $y=2\sqrt{13}$, $z=3\sqrt{13}$