# Solving Right Triangles

## Contents

#### Right Triangle Review

A right triangle is a triangle with one right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. The angles opposite the legs, by definition, are complementary. Suppose that the legs have lengths a and b , and the hypotenuse has length c . The Pythagorean Theorem states that in all right triangles, a 2 + b 2 = c 2 . For a more thorough discussion of right triangles, see Right Triangles.

In this text, we will label the vertices of every right triangle A , B , and C . The angles will be labeled according to the vertex at which they are located. The side opposite angle A will be labeled side a , the side opposite angle B will be labeled side b , and the side opposite angle C will be labeled side c . Angle C we will designate as the right angle, and thus, side c will always be the hypotenuse. Angle A will always have its vertex at the origin, and angle B will always have its vertex at the point (b, a) . Any right triangle can be situated on the coordinate axes to be in this position:

Figure %: A right triangle with vertex A at the origin and angle A in standard position
The triangle above is the general form of the right triangles we'll study in these sections on solving right triangles. Whenever you need to diagram a right triangle, this model is convenient and easy to follow.

In Trigonometic Functions, we defined the trigonometric functions using the coordinates of a point on the terminal side of an angle in standard position. With right triangles, we have a new way to define the trigonometric functions. Instead of using coordinates, we can use the lengths of certain sides of the triangle. These sides are the hypotenuse, the opposite side, and adjacent side. Using the figure above, the hypotenuse is side c , the opposite side is side a , and the adjacent side is side b . Here are the sides of a general right triangle labeled in the coordinate lane.

Figure %: The hypotenuse, opposite side, and adjacent side of a right triangle