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A triangle has sides with lengths of $5\,\text{inches}$ , $12\,\text{inches}$ , and $13\,\text{inches}$ . Is it a right triangle? $\newline$ Choices: $\newline$ (A) yes $\newline$ (B) no

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Converse of the Pythagorean theorem: is it a right triangle?

Full solution

Q. A triangle has sides with lengths of

5\,\text{inches}

,

12\,\text{inches}

, and

13\,\text{inches}

. Is it a right triangle?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Identify longest side: Identify the longest side to test for the hypotenuse. $\newline$ Given sides are $5$ inches, $12$ inches, and $13$ inches. The longest side is $13$ inches, which could be the hypotenuse.
Apply Pythagorean Theorem: Apply the Pythagorean Theorem to check if it's a right triangle. $\newline$ Using the formula $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse. $\newline$ Calculate $5^2 + 12^2$ and compare it to $13^2$ . $\newline$ $5^2 = 25$ , $12^2 = 144$ , $13^2 = 169$ . $\newline$ $25 + 144 = 169$ .
Compare side lengths: Compare the sum of squares of the two shorter sides with the square of the longest side. $\newline$ Since $169 = 169$ , the sides satisfy the Pythagorean Theorem.

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