Right triangles and the Pythagorean Theorem
A right triangle is a triangle with one right angle, that is, one angle that measures 90°. By
convention we usually call a right triangle ΔABC, and give the angles and sides the labels
shown in this picture. The vertex of the
right angle is labeled with the capital
letter C; the other vertices are labeled A
and B. The right angle is often marked
with a small square.
The lowercase letters c, a, and b are used
to represent the lengths of the sides of the
right triangle. The label of a vertex and
the label of the opposite side correspond. So the side opposite the right angle C is labeled
c; this special side is called the hypotenuse of the right triangle. The side opposite angle
A is labeled a; the side opposite angle B is labeled b. These other two sides (not the
hypotenuse) are called the legs of the right triangle.
The Pythagorean Theorem gives an important relationship among the sides of a right
triangle. This Theorem can be used to find the third side of a right triangle when two
sides are known.
Pythagorean Theorem: Suppose ΔABC is a right triangle with right angle C. Suppose c
represents the length of the hypotenuse, and a and b are the lengths of the legs. Then
222
bac
+=
.
Also, if
222
bac
+=
for any triangle ΔABC, then the triangle is a right triangle with right
angle C.
Example 1: Find the hypotenuse of a right triangle whose legs have lengths 5 inches and
12 inches.
If a picture isn’t given, draw one and
label what you know.
You want to find the hypotenuse, which
is labeled c. Use the Pythagorean
Theorem.
Substitute the values you know and
solve for c. In the square root step, we
take only the positive value since c is a
length.
feet13c
13169c
16914425c
125c
bac
2
2
222
222
=
==
=+=
+=
+=
Answers to odd exercises on reverse: 1. 10 feet; 3. 5
5
inches; 5. 10
3
feet; 7. 4
3
;
9. 8
3
; 11.
2
x ; 13. no, since
41251636 ;bac
222
=+≠+≠
