Question

given the following side lengths of a triangle, use the pythagorean theorem to determine whether the triangle is a right triangle.
a = 33 yd
b = 56 yd
c = 65 yd

show your work here
hint: to add the square root symbol (√□), type
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Explanation:

Step1: Recall Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here, we assume \(c\) is the longest side (hypotenuse candidate), so we check if \(33^{2}+56^{2}=65^{2}\).

Step2: Calculate \(a^{2}\)

\(a = 33\), so \(a^{2}=33\times33 = 1089\)

Step3: Calculate \(b^{2}\)

\(b = 56\), so \(b^{2}=56\times56=3136\)

Step4: Calculate \(a^{2}+b^{2}\)

\(a^{2}+b^{2}=1089 + 3136=4225\)

Step5: Calculate \(c^{2}\)

\(c = 65\), so \(c^{2}=65\times65 = 4225\)

Step6: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(a^{2}+b^{2}=4225\) and \(c^{2}=4225\), we have \(a^{2}+b^{2}=c^{2}\)

Answer:

The triangle with side lengths \(a = 33\) yd, \(b = 56\) yd, and \(c = 65\) yd is a right triangle because \(33^{2}+56^{2}=65^{2}\) (i.e., \(1089 + 3136=4225\) and \(65^{2}=4225\)).