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Special Triangles

Right Triangles



A triangle with one right angle is called a right triangle. The side opposite the right angle is called the hypotenuse of the triangle. The other two sides are called legs. The other two angles have no special name, but they are always complementary. Do you see why? The total angle sum of a triangle is 180 degrees, and the right angle is 90 degrees, so the other two must sum to 90 degrees.

Figure %: A right triangle
The triangle above has side c as its hypotenuse, sides a and b as its legs, and angle C as its right angle. Angles A and B are complementary.

There are two types of right triangles that every mathematician should know very well. One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles. The angles of the triangle will be 30, 60, and 90 degrees, giving the triangle its name: 30-60-90 triangle. The ratio of side lengths in such triangles is always the same: if the leg opposite the 30 degree angle is of length x , the leg opposite the 60 degree angle will be of x , and the hypotenuse across from the right angle will be 2x . Here is a 30-60-90 triangle pictured below.

Figure %: A 30-60-90 triangle

The other common right triangle results from the pair of triangles created when a diagonal divides a square into two triangles. Each of these triangles is congruent, and has angles of measures 45, 45, and 90 degrees. If the legs opposite the 45 degree angles are of length x , the hypotenuse has a length of x . This ratio holds true for all 45-45-90 triangles. 45-45-90 triangles are also often called isosceles right triangles.

Figure %: A 45-45-90 triangle

One last characteristic to note is that the legs of a right triangle are also altitudes of the triangle. Therefore, the area of a right triangle is one-half the product of the lengths of its legs.