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A triangle has sides with lengths of $9\,\text{miles}$ , $15\,\text{miles}$ , and $17\,\text{miles}$ . 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{miles}

,

15\,\text{miles}

, and

17\,\text{miles}

. Is it a right triangle?

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Identify sides and condition: Step $1$ : Identify the sides of the triangle and the condition for a right triangle. $\newline$ We have sides $9$ miles, $15$ miles, and $17$ miles. For a triangle to be a right triangle, the square of the longest side (hypotenuse) must equal the sum of the squares of the other two sides.
Calculate squares of sides: Step $2$ : Calculate the squares of the sides. $\newline$ $9^2 = 81$ $\newline$ $15^2 = 225$ $\newline$ $17^2 = 289$
Check sum of squares: Step $3$ : Check if the sum of the squares of the shorter sides equals the square of the longest side. $\newline$ $81 + 225 = 306$
Compare with longest side: Step $4$ : Compare the sum from Step $3$ with the square of the longest side. $306 \neq 289$

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