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A triangle has sides with lengths of $9\,\text{millimeters}$ , $14\,\text{millimeters}$ , and $17\,\text{millimeters}$ . 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

9\,\text{millimeters}

,

14\,\text{millimeters}

, and

17\,\text{millimeters}

. Is it a right triangle?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Identify longest side: Step $1$ : Identify the longest side to test for the hypotenuse. $\newline$ Given side lengths: $9\,\text{mm}$ , $14\,\text{mm}$ , $17\,\text{mm}$ . $\newline$ The longest side, which could be the hypotenuse, is $17\,\text{mm}$ .
Apply Pythagorean Theorem: Step $2$ : 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 $9^2 + 14^2$ and compare it to $17^2$ . $\newline$ $9^2 = 81$ , $14^2 = 196$ , $17^2 = 289$ . $\newline$ $81 + 196 = 277$ .
Compare side lengths: Step $3$ : Compare the sum of squares of the two shorter sides with the square of the longest side. $\newline$ $277$ does not equal $289$ . $\newline$ Since $277 \neq 289$ , the triangle is not a right triangle.

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