Question

1. a triangle has sides measuring 6 inches, 7 inches and 12 inches. is the triangle a right triangle? show or explain how you know.

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle, the Pythagorean theorem states that for side lengths \(a\), \(b\) (the two shorter sides) and \(c\) (the longest side), \(a^{2}+b^{2}=c^{2}\). Here, \(a = 6\), \(b = 7\) and \(c=12\).

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

Calculate \(a^{2}=6^{2}=36\) and \(b^{2}=7^{2} = 49\). Then \(a^{2}+b^{2}=36 + 49=85\).

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

Calculate \(c^{2}=12^{2}=144\).

Step4: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(85
eq144\) (i.e., \(a^{2}+b^{2}
eq c^{2}\)).

Answer:

The triangle with side lengths 6 inches, 7 inches, and 12 inches is not a right - triangle because \(6^{2}+7^{2}
eq12^{2}\) ( \(36 + 49=85\) and \(12^{2}=144\), and \(85
eq144\)).