Question

pythagorean theorem (radical answers)
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question
find the length of the third side. if necessary, write in simplest radical form.
(image of a right triangle with legs √7 and 3)
answer attempt 1 out of 2
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Explanation:

Step1: Identify the triangle type

The triangle is right - angled, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\) (if we are finding the hypotenuse) or \(b^{2}=c^{2}-a^{2}\) (if we are finding a leg). Here, the two legs are \(\sqrt{7}\) and \(3\), and we need to find the hypotenuse.
Let \(a = \sqrt{7}\), \(b = 3\), and \(c\) be the hypotenuse. Then by the Pythagorean theorem, \(c^{2}=(\sqrt{7})^{2}+3^{2}\).

Step2: Calculate the values

First, calculate \((\sqrt{7})^{2}=7\) and \(3^{2}=9\). Then \(c^{2}=7 + 9=16\).

Step3: Find the length of \(c\)

Take the square root of both sides. Since \(c\) represents the length of a side of a triangle, \(c>0\). So \(c=\sqrt{16} = 4\).

Answer:

\(4\)