Each side of right triangle DEF is an integer. Which of the following, if known, would yield exactly one area for triangle DEF?

[A] One of the sides is 3.

[B] One of the sides is 5.

[C] One of the sides is 10.

[D] The hypotenuse is 17.

[E] The perimeter of DEF is 60.

[F] The square root of the sum of the squares of the two shorter sides is equal to the longest side.

## Answer

This is a great question to test your knowledge of Pythagorean triplets: 3:4:5, 5:12:13, 8:15:17, and 7:24:25, to name the most important.

Let’s start first with [A]. If the shortest side is 3, there is only one possible triangle this could be: a 3:4:5. Note how none of the other Pythagorean triplets have a 3.

[B], however, could result in a 3:4:5 or a 5:12:13, so it is not the answer.

[C] can be a 6:8:10 (double a 3:4:5) or a 10:24:26 (double a 5:12:13). Not an answer.

[D] The only such triangle is a 8:15:17. Answer.

[E] Using a 3:4:5 lengths we can get a perimeter of 12. If multiply this by 5, we get a 15:20:25 triangle, which has a perimeter of 60. If we do something similar with a 5:12:13, we get a 10:24:26. Not an answer.

[F] This is another way of stating the Pythagorean theorem. Therefore, it describes any of the Pythagorean triplets. Not an answer.

Answers: [A] and [D]

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I think you misunderstand the [F], it first adds the two longer side, then takes square roots, not Pythagorean theorem, which first squares the sides then adds.

You are absolutely right–it looks like we made a mistake in our wording here! We will work on getting this fixed and updated. Thank you for pointing out our mistake and helping us to make our blog the best it can be 🙂