Question

1. find the value of x. then determine if the side lengths form a pythagorean triple.

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 7\), \(b=x\) and hypotenuse \(c = 9\), the Pythagorean theorem states that \(a^{2}+b^{2}=c^{2}\). Substituting the values, we get \(7^{2}+x^{2}=9^{2}\).

Step2: Solve for \(x^{2}\)

First, calculate \(7^{2}=49\) and \(9^{2}=81\). Then the equation becomes \(49 + x^{2}=81\). Subtract 49 from both sides: \(x^{2}=81 - 49=32\).

Step3: Solve for \(x\)

Take the square root of both sides. Since \(x\) represents a length, we take the positive square root. So \(x=\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\approx5.66\).

Step4: Check for Pythagorean triple

A Pythagorean triple consists of three positive integers \(a\), \(b\), \(c\) such that \(a^{2}+b^{2}=c^{2}\). Here, \(7\), \(4\sqrt{2}\) (or approximately \(5.66\)) and \(9\) are not all integers (because \(4\sqrt{2}\) is an irrational number), so they do not form a Pythagorean triple.

Answer:

The value of \(x\) is \(4\sqrt{2}\) (or approximately \(5.66\)) and the side lengths do not form a Pythagorean triple.