Question

unit 6 review right triangles and trigonometry
find the value of the variables.
x=
y=
z=
30
6

Explanation:

Step1: Identify similar triangles property

The large right triangle is split into two smaller similar right triangles, so we use geometric mean (altitude-on-hypotenuse) theorems.

Step2: Calculate hypotenuse segments

Total hypotenuse length: $30$, one segment is $6$, so the other segment is $30-6=24$.

Step3: Solve for $y$ (altitude)

$y$ is geometric mean of $6$ and $24$:
$$y = \sqrt{6 \times 24} = \sqrt{144} = 12$$

Step4: Solve for $z$ (short leg)

$z$ is geometric mean of $6$ and $30$:
$$z = \sqrt{6 \times 30} = \sqrt{180} = 6\sqrt{5}$$

Step5: Solve for $x$ (long leg)

$x$ is geometric mean of $24$ and $30$:
$$x = \sqrt{24 \times 30} = \sqrt{720} = 12\sqrt{5}$$

Answer:

$x=12\sqrt{5}$, $y=12$, $z=6\sqrt{5}$