Question

1. the green triangle is set inside a rectangle. is the green triangle a right triangle? explain.

\\(\sqrt{224}\\) in.
19 in.
\\(\sqrt{137}\\) in.

Explanation:

Step1: Recall Pythagorean theorem

For a triangle to be right - angled, the sum of the squares of the two shorter sides should equal the square of the longest side. Let the sides of the triangle be \(a = \sqrt{224}\), \(b=\sqrt{137}\) and \(c = 19\). We need to check if \(a^{2}+b^{2}=c^{2}\).

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

\(a=\sqrt{224}\), so \(a^{2}=(\sqrt{224})^{2}=224\)

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

\(b = \sqrt{137}\), so \(b^{2}=(\sqrt{137})^{2}=137\)

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

\(a^{2}+b^{2}=224 + 137=361\)

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

\(c = 19\), so \(c^{2}=19^{2}=361\)

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

Since \(a^{2}+b^{2}=361\) and \(c^{2}=361\), we have \(a^{2}+b^{2}=c^{2}\)

Answer:

Yes, the green triangle is a right triangle. Because when we check the Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\)) for the sides \(\sqrt{224}\), \(\sqrt{137}\) and \(19\), we find that \((\sqrt{224})^{2}+(\sqrt{137})^{2}=224 + 137=361\) and \(19^{2}=361\), so the Pythagorean theorem holds.