8th Grade - Special Right Triangles


  • Aside from the Pythagorean Theorem, there are helpful theorems we can use to solve for the measure of side lengths or angles of a right triangle.
  • Two special triangles are important in geometry - 30°-60°-90° Triangle, and 45°-45°-90° Triangle.
  • 45°-45°-90° Triangle is also called the Isosceles Right Triangle.

Theorems on Special Right Triangles

  • 30°-60°-90° Triangle Theorem - In 30°-60°-90° Triangle, the side opposite the 30° angle is half as long as the hypotenuse, and the side opposite the 60° angle is 3 times as long as the side opposite the 30° angle.

  • The converse of the 30°-60°-90° Triangle - If one leg is half as long as the hypotenuse in a right triangle, then the opposite angle has a measure of 30 degrees.
  • Isosceles Right Triangle Theorem - In an isosceles right triangle, the hypotenuse is 2 times as long as either of the legs. To illustrate, we have AC=BC with hypotenuse AB=x2.

Solved Examples

Example 1. Refer to the figure above. If c=30, what is the length of a and b?


Since the length of the hypotenuse is 30 units, the length of a is 15 units.

If a=15 units, then b=153 units.


Example 2. Refer to the figure above. If the length of leg b is 24 units, what are the lengths of the hypotenuse and the other leg?


ACB is an isosceles right triangle.

If one leg measures 24 units, the other leg is 24 units. 

The length of the hypotenuse is 242 units.

Other Theorems on Right Triangles

  • In a right triangle, the altitude to the hypotenuse divides the triangle into two similar triangles to the original triangle.
  • In any right triangle, the altitude to the hypotenuse is the geometric mean between the segments into which it divides the hypotenuse. To illustrate, we have the equation BDAD=ADCD.
  • Each leg is a geometric mean between the hypotenuse and the hypotenuse segment adjacent to the leg. To illustrate, we have the following equations: BCAB=ABBD and BCAC=ACCD

Cheat Sheet

  • In the 30-60-90 Triangle, if the longer leg a has a measure x and the hypotenuse is denoted by c, then c=2x and b=x3 units.
  • In the 45-45-90 Triangle, if one leg measures x, then the other leg measures x and the hypotenuse measures x2 units.
  • The side opposite the 30-degree angle is the shorter leg.
  • The side opposite the 60-degree angle is the longer leg.
  • The angle opposite the shorter leg measures 30°.
  • The angle opposite the longer leg measures 60°.

Blunder Areas

  • The sides of a right triangle are not closed using the symbols a, b, or c. Any letter symbol can be used as long as the concepts of the theorems are appropriately utilized.
  • It is typical that uppercase letters are used to denote angles while lowercase letters are used to denote side lengths of a right triangle.
  • 30-60-90 Triangle Theorem is used to derive the formula of the Area of an Equilateral Triangle, A=s234.
  • The Isosceles Triangle Theorem and Median Theorem of a right triangle can be used in proving the 30-60-90 Triangle Theorem.
  • Pythagorean Theorem can be used in proving the 45-45-90 Triangle Theorem.