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A triangle has sides with lengths of $3\,\text{km}$ , $4\,\text{km}$ , and $5\,\text{km}$ . 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

3\,\text{km}

,

4\,\text{km}

, and

5\,\text{km}

. Is it a right triangle?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Identify longest side: Identify the longest side to test for the right triangle using the Pythagorean theorem. $\newline$ Given sides: $3\,\text{km}$ , $4\,\text{km}$ , $5\,\text{km}$ . Longest side (potential hypotenuse) is $5\,\text{km}$ .
Apply Pythagorean theorem: Apply the Pythagorean theorem: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse. $\newline$ Calculation: $3^2 + 4^2 = 5^2$ $\newline$ $9 + 16 = 25$ $\newline$ $25 = 25$
Compare results: Compare the results. $\newline$ Since $25$ equals $25$ , the sides satisfy the Pythagorean theorem, confirming it's a right triangle.

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