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Section 9.6 The Pythagorean Theorem
959
Version: Fall 2007
9.6 The Pythagorean Theorem
Pythagoras.
Pythagoras was a Greek mathematician and philosopher, born on the island of Samos
(ca. 582 BC). He founded a number of schools, one in particular in a town in south-
ern Italy called Crotone, whose members eventually became
known as the Pythagoreans. The inner circle at the school,
the
Mathematikoi
, lived at the school, rid themselves of all
personal possessions, were vegetarians, and observed a strict
vow of silence. They studied mathematics, philosophy, and
music, and held the belief that numbers constitute the true
nature of things, giving numbers a mystical or even spiritual
quality.
Today, nothing is known of Pythagoras’s writings, per-
haps due to the secrecy and silence of the Pythagorean so-
ciety. However, one of the most famous theorems in all of
mathematics does bear his name, the
Pythagorean Theorem.
Pythagorean Theorem.
Let
c
represent the length of the
hypotenuse
, the side
of a right triangle directly opposite the right angle (a right angle measures 90
◦
) of
the triangle. The remaining sides of the right triangle are called the
legs
of the
right triangle, whose lengths are designated by the letters
a
and
b
.
a
b
c
The relationship involving the legs and hypotenuse of the right triangle, given by
a
2
+
b
2
=
c
2
,
(1)
is called the
Pythagorean Theorem
.
Note that the Pythagorean Theorem can only be applied to right triangles.
Let’s look at a simple application of the Pythagorean Theorem (
1
).
⚏
Example 2.
Given that the length of one leg of a right triangle is 4 centimeters
and the hypotenuse has length 8 centimeters, ﬁnd the length of the second leg.
Let’s begin by sketching and labeling a right triangle with the given information.
We will let
x
represent the length of the missing leg.
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Chapter 9 Radical Functions
Version: Fall 2007
x
4 cm
8 cm
Figure 1.
A sketch makes things a
bit easier.
Here is an important piece of advice.
Tip 3.
The hypotenuse is the longest side of the right triangle. It is located di-
rectly opposite the right angle of the triangle. Most importantly, it is the quantity
that is
isolated
by itself in the Pythagorean Theorem.
a
2
+
b
2
=
c
2
Always isolate the quantity representing the hypotenuse on one side of the equa-
tion. The legs go on the other side of the equation.
So, taking the tip to heart, and noting the lengths of the legs and hypotenuse in
Figure 1
, we write
4
2
+
x
2
= 8
2
.
Square, then isolate
x
on one side of the equation.
16 +
x
2
= 64
x
2
= 48
Normally, we would take plus or minus the square root in solving this equation, but
x
represents the length of a leg, which must be a positive number. Hence, we take just
the positive square root of 48.
x

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