Question

find the missing side length. keep answers in a simplified radical form when numbers. 15 mi x 16 mi x = miles do the side lengths form a pythagorean

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, the Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse (the longest side). Here, assume the sides of the right - triangle are $a = 15$, $b=x$, and $c = 16$. Then $x=\sqrt{c^{2}-a^{2}}$.
$x=\sqrt{16^{2}-15^{2}}$

Step2: Calculate the squares

First, calculate $16^{2}=256$ and $15^{2}=225$.
$x=\sqrt{256 - 225}$

Step3: Subtract and find square - root

$256-225 = 31$, so $x=\sqrt{31}$ miles.
To check if the side lengths form a Pythagorean triple, a Pythagorean triple consists of three positive integers $a$, $b$, and $c$ such that $a^{2}+b^{2}=c^{2}$. Since $\sqrt{31}$ is not an integer, the side lengths $15$, $\sqrt{31}$, and $16$ do not form a Pythagorean triple.

Answer:

$x=\sqrt{31}$; No, the side lengths do not form a Pythagorean triple.