How To Solve Special Right Triangles

In math and geometry, one of the skills that sets the experts apart from the pretenders is the knowledge of tricks and shortcuts. The time you spend learning them pays off in time saved when you solve problems. For example, it's worthwhile to know two special right triangles that, once you recognize them, are a snap to solve. The two triangles in particular are the 30-60-90 and the 45-45-90.

Triangles are three-sided polygons whose internal angles add up to 180 degrees. The right triangle is a special case in which one of the angles is 90 degrees, so the other two angles by definition must add up to 90. The sine, cosine, tangent and other trigonometric functions provide ways to calculate the internal angles of right triangles as well as the length of their sides. Another indispensable calculating tool for right triangles is the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, or

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

When you're working on any kind of right triangle problem, you're usually given at least one angle and one side and asked to calculate the remaining angles and sides. Using the Pythagorean formula above, you can calculate the length of any side if you're given the other two. A big advantage of the special right triangles is that the proportions of the lengths of their sides are always the same, so you can find the length of all sides if you're given only one. Also, if you're given only one side, and the triangle is special, you can find the values of the angles as well.

As the name implies, the 30-60-90 right triangle has internal angles of 30, 60 and 90 degrees. As a consequence, the sides of this triangle fall into the proportions, 1: 2: √3, where 1 and √3 are the lengths of the opposite and adjacent sides and 2 is the hypotenuse. These numbers always go together: if you solve the sides of a right triangle and find they fit the pattern, 1, 2, √3, you know the angles will be 30, 60 and 90 degrees. Likewise, if you're given one of the angles as 30, you know the other two are 60 and 90, and also that the sides will have the proportions, 1: 2: √3.

The 45-45-90 triangle works much like the 30-60-90, except that two angles are equal, as are the opposite and adjacent sides. It has internal angles of 45, 45 and 90 degrees. The proportions of the sides of the triangle are 1: 1: √2, with the proportion of the hypotenuse being √2. The other two sides are equal in length to each other. If you're working on a right triangle and one of the internal angles is 45 degrees, you know in an instant that the remaining angle must also be 45 degrees, because the whole triangle must add up to 180 degrees.

When solving the two special right triangles, keep in mind that it's the ​proportions​ of the sides that matter, not their measurement in absolute terms. For example, a triangle has sides that measure 1 foot, and 1 foot, and √2 feet, so you know it's a 45-45-90 triangle and has internal angles of 45, 45, and 90 degrees.

But what do you do with a right triangle whose sides measure √17 feet and √17 feet? The proportions of the sides are the key. Since the two sides are identical, the proportion is 1:1 with one another, and because it's a right triangle, the proportion of the hypotenuse is 1:√2 with either of the other sides. The equal proportions tip you off that the sides are 1, 1, √2, which belongs only to the 45-45-90 special triangle. To find the hypotenuse, multiply √17 by √2 to get √34 feet.