12.6
The Pythagorean Theorem
and Its Converse
Goal
Use the Pythagorean
theorem and its converse
What is the distance from home plate to second base?
You will use the Pythagorean theorem
in Exercise 31 to find the distance from
home plate to second base of a standard
baseball diamond.
Key Words
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•
•
•
•
theorem
Pythagorean theorem
hypotenuse
legs of a right triangle
converse
A theorem is a statement that can be proven to be true. The Pythagorean
theorem states a relationship among the sides of a right triangle. The hypotenuse
is the side opposite the right angle. The other two sides are the legs.
THE PYTHAGOREAN THEOREM
If a triangle is a right triangle, then the
sum of the squares of the lengths of
the legs a and b equals the square of
the length of the hypotenuse c.
a b c
2
2
c
a
b
2
In Exercise 23 in Lesson 12.9, you will outline a proof of the Pythagorean theorem.
Student Help
STUDY TIP
When you use the
Pythagorean theorem
to find the length of a
side of a right triangle,
you need only the
positive square root
because the length of
a side cannot be
negative.
EXAMPLE
1
Use the Pythagorean Theorem
a. Given a 6 and b 8, find c. Use the Pythagorean theorem: a2 b 2 c 2.
62 82 c 2
100 c 2
c
a6
100 c
2
10 c
b8
b. Given a 5 and c 6, find b. Use the Pythagorean theorem: a2 b 2 c 2.
52 b2 62
b2 62 52
a5
b
b 11
2
b 11 ≈ 3.32
724
Chapter 12.6 Radicals and More Connections to Geometry
c6