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A triangle has sides with lengths of $12\,\text{miles}$ , $16\,\text{miles}$ , and $20\,\text{miles}$ . 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

12\,\text{miles}

,

16\,\text{miles}

, and

20\,\text{miles}

. 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: $12$ miles, $16$ miles, $20$ miles. $\newline$ Longest side (hypotenuse candidate): $20$ miles. $\newline$ Other sides (legs): $12$ miles and $16$ miles.
Apply Pythagorean Theorem: Apply the Pythagorean theorem to check if it's a right triangle. $\newline$ Using the formula: $a^2 + b^2 = c^2$ $\newline$ Calculate: $12^2 + 16^2$ and compare it to $20^2$ . $\newline$ $12^2 = 144$ , $16^2 = 256$ , $20^2 = 400$ . $\newline$ Sum of squares of legs: $144 + 256 = 400$ .
Compare Squares: Compare the sum of the squares of the legs to the square of the longest side. $\newline$ Calculated sum of squares of legs: $400$ . $\newline$ Square of the longest side (hypotenuse): $400$ . $\newline$ Since $144 + 256 = 400$ equals $20^2$ , the sides satisfy the Pythagorean theorem.

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