Question

1. the research triangle in north carolina is formed by raleigh, durham, and chapel hill. is this triangle a right triangle? explain.

Explanation:

Step1: Recall the Pythagorean theorem

For a triangle with side lengths \(a\), \(b\), \(c\) (\(c\) is the longest side), if \(a^{2}+b^{2}=c^{2}\), then the triangle is a right - triangle. Let \(a = 12\), \(b=24\), \(c = 29\).

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

• Calculate \(a^{2}+b^{2}\):

\(a^{2}+b^{2}=12^{2}+24^{2}=144 + 576=720\)

• Calculate \(c^{2}\):

\(c^{2}=29^{2}=841\)

Step3: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(720
eq841\) (i.e., \(12^{2}+24^{2}
eq29^{2}\))

Answer:

No, the triangle formed by Raleigh, Durham, and Chapel Hill is not a right - triangle because \(12^{2}+24^{2}=144 + 576 = 720\) and \(29^{2}=841\), and \(720
eq841\), so the Pythagorean theorem is not satisfied.