Question

for the following right triangle, find the side length x.

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). Here, the two legs are \(a = 8\) and \(b=15\), and the hypotenuse is \(x\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 8\) and \(b = 15\) into the formula \(a^{2}+b^{2}=c^{2}\) (where \(c=x\)).
We get \(8^{2}+15^{2}=x^{2}\).
Calculate \(8^{2}=64\) and \(15^{2}=225\). Then \(64 + 225=x^{2}\).
\(64+225 = 289\), so \(x^{2}=289\).

Step3: Solve for \(x\)

Take the square root of both sides. Since \(x\) represents the length of a side of a triangle, we take the positive square root. \(\sqrt{x^{2}}=\sqrt{289}\), so \(x = 17\).

Answer:

\(x = 17\)