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A triangle has sides with lengths of $2\,\text{km}$ , $8\,\text{km}$ , and $9\,\text{km}$ . 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

2\,\text{km}

,

8\,\text{km}

, and

9\,\text{km}

. 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 right triangle properties. $\newline$ We know the sides are $2\,\text{km}$ , $8\,\text{km}$ , and $9\,\text{km}$ . The longest side, which could be the hypotenuse if this is a right triangle, is $9\,\text{km}$ .
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 ( $9$ km), and $a$ and $b$ are the other two sides ( $2$ km and $8$ km). $\newline$ Calculate $2^2 + 8^2 = 4 + 64 = 68$ .
Compare Calculated Value: Step $3$ : Compare the calculated value with $c^2$ . $\newline$ $9^2 = 81$ . $\newline$ Since $68 \neq 81$ , the sides do not satisfy the Pythagorean Theorem.

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