Question

finding side lengths find the unknown side length. simplify answers that are radicals. tell whether the side lengths form a pythagorean triple.
13.
a right triangle with one leg 8, hypotenuse 16, and the other leg x (the right angle is between the leg of length 8 and the leg of length x)

Explanation:

Step1: Identify Pythagorean theorem parts

The triangle is right-angled, so use $a^2 + b^2 = c^2$. Here, $a=8$, $c=16$, $b=x$.

Step2: Rearrange to solve for $x$

Isolate $x^2$: $x^2 = c^2 - a^2$
Substitute values: $x^2 = 16^2 - 8^2$

Step3: Calculate the squares

$16^2=256$, $8^2=64$, so $x^2=256-64=192$

Step4: Simplify the radical

$x=\sqrt{192}=\sqrt{64\times3}=8\sqrt{3}$

Step5: Check Pythagorean triple

A triple requires all sides to be positive integers. $8\sqrt{3}$ is not an integer, so it is not a triple.

Answer:

The unknown side length is $8\sqrt{3}$. The side lengths do not form a Pythagorean triple.