Question

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 )`?

Answer 1

`24,33,42` are not the sides of a right-angled triangle.

Explanation:

Test each set of lengths using Pythagoras' Theorem.

Is `a^2 + b^2 = c^2?`

`36^2 +77^2 = 7225" " and 85^2 = 7225`
This is a right-angled triangle.

`20^2 +99^2 = 10,201" "and 101^2 =10,201`
This is a right-angled triangle.

`27^2 +120^2 = 15,129" "and 123^2 = 15,129`
This is a right-angled triangle.

`24^2 +33^2 = 1665" "and 42^2 = 1764`
This is not a right-angled triangle.

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

`24` is even, its square is even.
`33` is odd, its square is odd.

An odd number plus an even will always give an odd answer, but the square of `42` will be an even number.

Therefore `24^2 +33^2 != 42^2`

Answer 2

These lengths are not the side lengths of a right triangle:
`(24, 33, 42)`

But these sets actually are the side lengths of right triangles
`(36, 77, 85)`
`(20, 99, 101)`
`(27, 1 20, 123)`

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   `36^2 +   77^2`  equal   `85^2?`
Does `1296 + 5929` equal `7225?`
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  `24^2 +   33^2`  equal   `42^2?`
Does `576 + 1089`  equal `1764?`
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.
.................

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 `a, b,` and `c`
where `a^2 + b^2 = c^2` `larr` the formula for the sides of a right triangle

`a` is always the smallest number and `c` is always the largest, making `c` the length of the hypotenuse of a right triangle

Triples are written `(a, b, c)`,
A famous example of a Pythagorean triple is `(3, 4, 5)`
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 `(a, b, c)` is a right triangle, then all the multiples of that set are also right triangles. `(3, 4, 5)` is a right triangle, so `(6, 8, 10)` is too.
`3^2 + 4^2 = 5^2` and `6^2 + 8^2 = 10^2`

The side lengths of most right triangles are not Pythagorean triples. For example, `(1, 1, sqrt(2))` is a right triangle, but it's not a Pythagorean triple because its sides are 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 `(a, b, c)` sets on this list.
https://en.wikipedia.org/wiki/Pythagorean_triple#Examples
The triples are listed in numerical order by the length of the hypotenuse.
........................

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.