Right Triangle Basics

What Makes a Triangle Right?

A right triangle is a triangle with one very special property: one of its three angles is a perfect 90-degree angle. This is also known as a right angle. You can think of it as the corner of a square or a book.

Because the three angles in any triangle must add up to 180 degrees, the other two angles in a right triangle must be acute, meaning they are both less than 90 degrees. This simple 90-degree corner gives right triangles unique and useful properties that we can rely on.

The Sides of a Right Triangle

The sides of a right triangle have special names. It's important to know them, as they're the foundation for understanding how these triangles work.

The longest side, which is always found opposite the right angle, is called the hypotenuse.

hypotenuse

noun

The longest side of a right triangle, opposite the right angle.

The other two sides are called legs, but they also have more specific names depending on which angle you're focusing on. Relative to one of the acute angles, one leg is opposite (across from the angle) and the other is adjacent (next to the angle).

The hypotenuse is always the hypotenuse. The opposite and adjacent sides, however, depend entirely on your point of view—which acute angle you are looking at.

The Pythagorean Theorem

Ancient mathematicians discovered a remarkable relationship between the sides of a right triangle. This rule, known as the Pythagorean theorem, allows us to find the length of a missing side if we know the lengths of the other two.

The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle.

If we label the two legs 'a' and 'b' and the hypotenuse 'c', the theorem can be written as a simple formula.

a^2 + b^2 = c^2

Let's see how it works. Imagine a right triangle with legs of length 5 and 12. To find the hypotenuse, we plug these values into the theorem.

\begin{aligned}a^2 + b^2 &= c^2 \\ 5^2 + 12^2 &= c^2 \\ 25 + 144 &= c^2 \\ 169 &= c^2 \\ \sqrt{169} &= c \\ 13 &= c\end{aligned}

So, the hypotenuse is 13 units long.

The theorem also works in reverse. If you know the hypotenuse is 10 and one leg is 6, you can find the other leg.

\begin{aligned}a^2 + b^2 &= c^2 \\ 6^2 + b^2 &= 10^2 \\ 36 + b^2 &= 100 \\ b^2 &= 100 - 36 \\ b^2 &= 64 \\ b &= \sqrt{64} \\ b &= 8\end{aligned}

The missing leg is 8 units long. This powerful formula is a fundamental tool for working with right triangles.

Time to check your understanding.

Quiz Questions 1/6

What is the defining characteristic of a right triangle?

Quiz Questions 2/6

In a right triangle, what is the special name for the side opposite the 90-degree angle?

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