What is one way to identify a right triangle?

One effective way to identify if a triangle is a right triangle is by using the Pythagorean theorem.

The Pythagorean Theorem in Action

The Pythagorean theorem describes the relationship between the lengths of the sides of a right triangle. If a triangle is a right triangle, the square of the length of the longest side (called the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (called the legs).

Conversely, if the lengths of the sides of any triangle satisfy this specific relationship, then the triangle must be a right triangle. This is the principle used for identification.

As stated in the reference:

"if it's a right triangle. Then the sum of the squares of the legs adds up to the square of the hypotenuse."

This means if a triangle has sides with lengths a, b, and c, and the relationship $a^2 + b^2 = c^2$ holds true (where c is the longest side), then you can confidently identify it as a right triangle.

How to Apply the Method

To check if a triangle is a right triangle using the Pythagorean theorem, follow these steps:

1. Identify the longest side: This will be your potential hypotenuse (c).
2. Identify the two shorter sides: These will be your potential legs (a and b).
3. Square the lengths of all three sides ($a^2$, $b^2$, and $c^2$).
4. Add the squares of the two shorter sides ($a^2 + b^2$).
5. Compare the sum to the square of the longest side ($c^2$).

If...	Then the triangle is a...
$a^2 + b^2 = c^2$	Right Triangle
$a^2 + b^2 \neq c^2$	Not a Right Triangle

Example

Let's say you have a triangle with side lengths 3, 4, and 5.

1. Longest side (c) = 5
2. Shorter sides (a and b) = 3 and 4
3. Square the lengths:
   • $a^2 = 3^2 = 9$
   • $b^2 = 4^2 = 16$
   • $c^2 = 5^2 = 25$
4. Add the squares of the shorter sides: $a^2 + b^2 = 9 + 16 = 25$
5. Compare the sum to the square of the longest side: $25 = 25$.

Since $a^2 + b^2 = c^2$ (specifically, $3^2 + 4^2 = 5^2$), the triangle with sides 3, 4, and 5 is indeed a right triangle.

This method provides a definitive test based purely on the side lengths.