Question

question
find the length of the third side. if necessary, write in simplest radical form.
answer
attempt 1 out of 3
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Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\). Let \(a = \sqrt{130}\), \(b\) be the unknown side, and \(c = 7\). Wait, no, correction: if the two legs of a right - triangle are \(a\) and \(b\), and the hypotenuse is \(c\), then \(a^{2}+b^{2}=c^{2}\). Here, assume the two given sides are \(a=\sqrt{130}\) and \(b\), and \(c\) is the hypotenuse. But if we consider the formula correctly, if one leg \(a=\sqrt{130}\) and the hypotenuse \(c = 7\) (no, that's wrong. Wait, actually, let's start over.

Let the two legs of the right - triangle be \(x\) and \(\sqrt{130}\), and the hypotenuse \(y\). By Pythagorean theorem \(x^{2}+(\sqrt{130})^{2}=y^{2}\). Wait, no, the problem is to find the third side. Let's assume the two sides are \(a=\sqrt{130}\) and \(b\), and we need to find the other side. If it's a right - triangle, and we know two sides. Let's assume the formula \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse.

Let the two sides be \(a = \sqrt{130}\) and \(b\), and assume the missing side is the hypotenuse. Then \(c^{2}=(\sqrt{130})^{2}+7^{2}\).

Step2: Calculate the value

\(c^{2}=130 + 49\).
\(c^{2}=179\).
\(c=\sqrt{179}\).

Another case: if the given side \(7\) is the hypotenuse and \(\sqrt{130}\) is a leg. Then \(7^{2}=(\sqrt{130})^{2}+x^{2}\), \(49 = 130+x^{2}\), \(x^{2}=49 - 130=- 81\) (not possible since the square of a real number cannot be negative). So the missing side is the hypotenuse.

Answer:

\(\sqrt{179}\)