Question

find the length of the third side. if necessary, write in simplest radical form. answer attempt 2 out of 2 submit answer

Explanation:

Step1: Identify the theorem

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) for a right - triangle, where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs. Here, we assume the hypotenuse is \(3\) and one leg is \(\sqrt{7}\), and we want to find the other leg. Let the unknown side be \(x\).

Step2: Apply the Pythagorean theorem

We have \(x^{2}+(\sqrt{7})^{2}=3^{2}\), which simplifies to \(x^{2}+7 = 9\).

Step3: Solve for \(x^{2}\)

Subtract 7 from both sides: \(x^{2}=9 - 7=2\).

Step4: 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. So \(x=\sqrt{2}\).

Answer:

\(\sqrt{2}\)