Question

question
find the length of the third side. if necessary, write in simplest radical form.
(there is a right triangle with legs 2 and 5, and the right angle between them. the hypotenuse is the side to find.)
answer attempt 1 out of 2
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Explanation:

Step1: Identify the triangle type

The triangle 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 \(c\), \(c^{2}=a^{2}+b^{2}\) (if we are finding the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if we are finding a leg). In this triangle, the two legs are of lengths \(a = 2\) and \(b=5\), and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Using the formula \(c=\sqrt{a^{2}+b^{2}}\), substitute \(a = 2\) and \(b = 5\) into the formula.
First, calculate \(a^{2}\) and \(b^{2}\): \(a^{2}=2^{2}=4\) and \(b^{2}=5^{2}=25\).
Then, find the sum of \(a^{2}\) and \(b^{2}\): \(a^{2}+b^{2}=4 + 25=29\).
So, \(c=\sqrt{29}\).

Answer:

\(\sqrt{29}\)