Question

find the length of the third side. if necessary, write in simplest radical form. √58 3 answer attempt 1 out of 2 submit answer

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 hypotenuse \(c\) and legs \(a\) and \(b\), \(a^{2}+b^{2}=c^{2}\). Let the unknown leg be \(x\), the given leg \(a = 3\), and the hypotenuse \(c=\sqrt{58}\).

Step2: Apply the Pythagorean theorem

We want to find the length of the unknown leg. Rearranging the Pythagorean theorem to solve for \(x\) (where \(x\) is one of the legs), we get \(x^{2}=c^{2}-a^{2}\). Substitute \(c = \sqrt{58}\) and \(a = 3\) into the formula:
\(x^{2}=(\sqrt{58})^{2}-3^{2}\)
First, calculate \((\sqrt{58})^{2}=58\) and \(3^{2} = 9\). Then \(x^{2}=58 - 9=49\).

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. So \(x=\sqrt{49} = 7\).

Answer:

\(7\)