# Which of these set of lengths are not the side lengths of a right triangle: #(36, 77, 85), (20, 99, 101 ), (27, 120, 123) and (24, 33, 42 )#?

##### 2 Answers

#### Explanation:

Test each set of lengths using Pythagoras' Theorem.

Is

This is a right-angled triangle.

This is a right-angled triangle.

This is a right-angled triangle.

This is not a right-angled triangle.

We could have seen that the last one would not work without doing any working:

An odd number plus an even will always give an odd answer, but the square of

Therefore

These lengths are not the side lengths of a right triangle:

But these sets actually are the side lengths of right triangles

#### Explanation:

To test each set, square all three numbers and see if the sum of the first two squares equals the sum of the third square.

Examples:

Does

Does

Since the squares of the two smaller sides do add up to the square of the longest side, then these are side lengths of a right triangle.

Does

Does

But in this case, the squares of the two smaller sides do not add up to the square of the longest side, so these are not the side lengths of a right triangle.

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Because all the answer choices are **whole numbers,** this is a question about "*Pythagorean triples*."

Here is what Wiki says about Pythagorean triples:

https://en.wikipedia.org/wiki/Pythagorean_triple

A Pythagorean triple

consists of three **whole numbers**

where *the formula for the sides of a right triangle*

Triples are written

A famous example of a Pythagorean triple is

*This means that the two regular sides of the triangle are 3 units and 4 units long, and the hypotenuse is 5 units long.*

If

The side lengths of most right triangles are not Pythagorean triples. For example, **not whole numbers.**

The oldest known record of a Pythagorean triple comes from a Babylonian clay tablet from about 1800 BC.

..................

I would never waste my study time squaring and adding

twelve big numbers.

Instead, just look up the given

https://en.wikipedia.org/wiki/Pythagorean_triple#Examples

The triples are listed in numerical order by the length of the hypotenuse.

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Standardized timed tests like the SAT, ACT, and GRE use Pythagorean triples all the time. If you know the sides by memory, you have a big advantage because you can avoid burning up your minutes on squaring.

Therefore, students should memorize the first three triples and be at least familiar with one or two more.