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

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

Take a Study Break!