Right triangles

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.

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

\(a^2+b^2=c^2\)

\(\sqrt{s^2+b^2}\)

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 \(c^2 < a^2 + b^2\), then the triangle is acute
  
  
• If \(c^2 = a^2 + b^2\), then the triangle is right
  
  
• If \(c^2 > a^2 + b^2\), then the triangle is obtuse

wo 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 \(b=a·\sqrt{3}\)

The hypotenuse is twice the length of the side \(a\) opposite the \(30°\) angle.

• Therefore, \(a\) must be the following \(a = c/2\)
• And \(b\) must be \(b=c/2·\sqrt{3}\)

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 \(\displaystyle \frac{1}{\sqrt{2}}\)
• This results in \(\displaystyle a=\frac{c}{\sqrt{2}}\)