How to Solve Special Right Triangles

By Isaiah David

There are two main types of special right triangles: the isosceles right triangle and the 30-60-90 triangle. Solving them can mean finding the value of a side, an angle or multiple sides and angles. By understanding the ratios of the sides to each other and to the angles, you can learn to solve either of these triangles with ease.

The Basics

Learn what a right triangle is. A right triangle any triangle with a right angle as one of its three angles. A right angle is one that measures exactly 90 degrees.

Learn what a special right triangle is. There are basically two special right triangles: the 30-60-90 right triangle and the 45-45-90, or isosceles, right triangle. The triangles are named for the measures of their three angles in degrees. The isosceles right triangle is also special because it has two sides which are the same.

Learn the parts of a right triangle. Each right triangle has two normal sides and a hypotenuse. The hypotenuse is the side opposite the right angle and is always the biggest side. The other sides are proportional to the angle opposite them. For example, in a 30-60-90 triangle, the side opposite the 30-degree angle is the smallest.

Learn the symbols for equality of angles and sides. Angles with a little curved line in them (drawn as in the illustration) are equal. Sides with a straight line through them have equal length. Angles with a right angle drawn inside (as in the illustration) are equal.

Solving an Isosceles Right Triangle

Know the ratio of the sides to the hypotenuse. The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the square of the other two sides. Therefore, if the two sides have the same value (we'll say that each sides has a length of 1), then the hypotenuse has a value of the square root of 2.

Understand that this ratio holds for any size of triangle. The ratio of any hypotenuse to a side is sqrt2 to 1. Therefore, a right triangle with a side of length 3 would have a hypotenuse of length 3sqrt2.

Learn to recognize an isosceles right triangle. To know that a triangle is an isosceles right triangle, you need to know that it is a right angle and that either the other two sides are equal or that the other two angles are equal. Either will do, since equal sides means equal angles and vice versa.

Learn to derive the hypotenuse from one of the sides. If you know that one side equals 5, for example, and you need to derive the hypotenuse, simply multiply it by sqrt2 to get 5sqrt2.

Learn to solve for a side when you know the hypotenuse. Divide the hypotenuse by the square root of 2 to get the value of either side. For example, if the hypotenuse is 4sqrt2, you get 4sqrt2/sqrt2 = 4therefore, both sides have a length of 4.

Solving a 30-60-90 Triangle

Learn the ratio of the sides of a 30-60-90 triangle. Unfortunately, you can't recognize a 30-60-90 special right triangle by anything as simple as equal sides and equal angles. But you can recognize it by the ratio of the three sides. The ratio of the 30 side to the 60 side to the hypotenuse is 1 to SQRT3 to 2.

Learn to recognize a 30-60-90 triangle based on the ratio of two sides. If you know that you have a right triangle, and the ratio of the larger side over the smaller is sqrt3 to 1, you have a 30-60-90 triangle. Then, you can fill in the values of the corresponding angles. For example, if the longer side is 2sqrt3 and the shorter is 2, you get:2sqrt3/2 = sqrt3/1.Therefore, the angle opposite the 2 is 30 degrees, and the angle opposite 2sqrt3 is 60 degrees.

Learn to solve based on the ratio of the hypotenuse to one of the sides. If the ratio of the hypotenuse to a side is either 2 to 1 (shorter side) or 2 to sqrt3 (longer side), then you have a 30-60-90 triangle and can fill in the angles.

Fill in the sides based on the angles and the value of one side. Sometimes all you will know is the value of one side and that it is a 30-60-90 triangle. In that case, you can fill in the other sides based on the ratio of the three sides. For example, if you know that you have a 30-60-90 triangle with a short side of 5, you know that the hypotenuse is twice the value, or 10. You also know that the longer side is sqrt3 times the value, or 5sqrt3.