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A triangle has sides with lengths of $8\,\text{inches}$ , $15\,\text{inches}$ , and $19\,\text{inches}$ . 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

8\,\text{inches}

,

15\,\text{inches}

, and

19\,\text{inches}

. Is it a right triangle?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Check Triangle Type: We need to check if the triangle with sides $8$ , $15$ , and $19$ inches is a right triangle using the Pythagorean Theorem. The longest side, $19$ inches, would be the hypotenuse if this is a right triangle. $\newline$ Calculation: $8^2 + 15^2$ should equal $19^2$ if it's a right triangle.
Calculate $8$ ^ $2$ : Calculate $8^2$ : $\newline$ $8 \times 8 = 64$ .
Calculate $15$ ^ $2$ : Calculate $15^2$ : $\newline$ $15 \times 15 = 225$ .
Add Shorter Sides: Add the squares of the two shorter sides: $\newline$ $64 + 225 = 289$ .
Calculate $19$ ^ $2$ : Calculate $19^2$ : $\newline$ $19 \times 19 = 361$ .
Compare Side Lengths: Compare the sum of the squares of the two shorter sides to the square of the longest side: $\newline$ $289$ does not equal $361$ . $\newline$ Since they are not equal, the triangle is not a right triangle.

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