RedCrab Math Tutorial

# Right Triangles And Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. The hypotenuse is the longest side of a right triangle. It is always across from the right angle.

Based on this right triangle, you can write an equation using the Pythagorean Theorem as follows:

• a2 + b2 = c2

## Pythagorean triple

A Pythagorean triplet is made up of three natural numbers, which may appear as lengths of the sides of a right triangle.
• A Pythagorean triple is a right triangle whose sides are in the ratio 3:4:5
• The side lengths do not need to measure 3, 4, and 5; however, they do need to reduce to that ratio
• For example, the side lengths 9, 12, and 15 are a Pythagorean triple because it simplifies to the ratio 3:4:5

## The Converse of the Pythagorean Theorem

You can also use the Pythagorean Theorem to determine if a triangle is acute, right, or obtuse. This is known as the converse of the Pythagorean Theorem, which reads as follows:
• If c2 < a2+ b2, then the triangle is acute
• If c2 = a2 + b2, then the triangle is right
• If c2 > a2 + b2, then the triangle is obtuse

## Special Right Triangles

Two types of right triangles are considered special right triangles. One of the special right triangles has angles that measure 30°, 60°, and 90°. The other special right triangle has angles that measure 45°, 45°, and 90°. The size of the triangle does not matter; it just needs to have specific measures for its angles.

## 30°-60°-90° Triangle

The lengths of the sides of a 30°-60°-90° triangle are in a ratio of
The hypotenuse is twice the length of the side opposite the30° angle.
• Therefore, a must be the following: a = c/2
• And b must be:
Using this information, you can determine the value of a and b for the triangle, if you know the value of c.

## 45°-45°-90° Triangle

The sides a of a 45° - 45° - 90° triangle are the same length.
• The ratio of side lengths to hypotenuse is
• This results in