6-3 The Pythagorean Theorem
Vocabulary
Pythagorean Theorem
Learn to use the Pythagorean Theorem and
its converse to solve problems.
leg
hypotenuse
REVIEW…
Additional Example: 1 Finding the Positive and Negative
Square Roots of a Number
Find the two square roots of each number.
A. 49
Taking the square root of a number is the inverse of
squaring the number.
62 = 36
–
This is a right triangle:
= 7 7 is a square root, since 7 • 7 = 49.
= –7 –7 is also a square root, since –7 • –7 = 49.
B. 100
36 = 6
Every positive number has two square roots, one positive and
one negative. One square root of 16 is 4, since 4 • 4 = 16.
The other square root of 16 is –4, since (–4) • (–4) is also 16.
You can write the square root of 16 as ±4, meaning “plus or
minus” 4.
49
49
100 = 10 10 is a square root, since 10 • 10 = 100.
– 100 = –10
–10 is also a square root, since –10 • –10 = 100.
C. 225
–
225
= 15 15 is a square root, since 15 • 15 = 225.
225
= –15
–15 is also a square root, since –15 • –15 = 225.
We call it a right triangle
because it contains a
right angle.
1
The measure of a right
angle is 90o
The little square in the
angle tells you it is a
right angle.
90o
90o
About 2,500 years ago, a
Greek mathematician named
Pythagorus discovered a
special relationship between
the sides of right triangles.
Pythagorus realized that if
you have a right triangle,
5
3
4
and you square the lengths
of the two sides that make
up the right angle,
and add them together,
5
5
3
3
4
32 42
4
32 + 42
2
you get the same number
you would get by squaring
the other side.
Is that correct?
?
32 + 42 = 52
?
5
9 + 16 = 25
3
4
32 + 42 = 52
The two sides which
come together in a right
angle are called
It is. And it is true for any
right triangle.
62 + 82 = 102
10
8
36 + 64 = 100
6
The two sides which
come together in a right
angle are called
The two sides which
come together in a right
angle are called
3
The lengths of the legs are
usually called a and b.
a
The side across from the
right angle is called the
a
b
And the length of the
hypotenuse
is usually labeled c.
a
c
b
The relationship Pythagorus
discovered is now called
The Pythagorean Theorem:
a
b
The Pythagorean Theorem
says, given the right triangle
with legs a and b and
hypotenuse c,
a
c
b
c
b
then a 2 + b 2 = c 2 .
a
c
b
4
Find the length of a diagonal
of the rectangle:
Find the length of a diagonal
of the rectangle:
15"
?
15"
8"
?
c
b=8
8"
a = 15
152 + 82 = c 2
a 2 + b2 = c 2
225 + 64 = c 2
c 2 = 289
b=8
Find the length of a diagonal
of the rectangle:
c = 17
15"
c
8"
17
a = 15
Practice using
The Pythagorean Theorem
to solve these right triangles:
c = 13
5
12
5
b = 24
b
10
26
a 2 + b2 = c 2
102 + b 2 = 262
100 + b 2 = 676
b 2 = 676 − 100
b 2 = 576
10 (a)
26
(c)
b = 24
12
b= 9
15
a2 + b2 = c2
Additional Example 1A: Find the the Length of a
Hypotenuse
Find the length of the hypotenuse.
A.
c
4
Additional Example 1B: Find the the Length of a
Hypotenuse
Find the length of the hypotenuse.
B.
triangle with coordinates
(1, –2), (1, 7), and (13, –2)
5
a2 + b2 = c2
42 + 52 = c2
16 + 25 = c2
41 = c2
41 = c
6.40 c
Pythagorean Theorem
Substitute for a and b.
Simplify powers.
Solve for c; c = c2.
a2 + b2 = c2
92 + 122 = c2
81 + 141 = c2
225 = c
15 = c
Pythagorean Theorem
Substitute for a and b.
Simplify powers.
Solve for c; c = c2.
6
Try This: Example 1A
Find the length of the hypotenuse.
A.
Find the length of the hypotenuse.
B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2)
c
5
Try This: Example 1B
y
(–2, 4)
7
The points form a right triangle.
a2 + b2 = c2
52 + 72 = c2
25 + 49 = c2
74 = c
8.60 c
Pythagorean Theorem
Substitute for a and b.
Simplify powers.
Solve for c; c = c2.
x
(3, –2)
(–2, –2)
Additional Example: 2 Finding the Length of a Leg in a
Right Triangle
Solve for the unknown side in the right triangle.
a2 + b2 = c2
b 72 + b2 = 252
49 + b2 = 625
–49
–49
b2 = 576
25
7
b = 24
Pythagorean Theorem
Substitute for a and c.
Simplify powers.
6
6
a
4
A = 1hb =
2
a2 + b2 = c2
a2 + 42 = 62
a2 + 16 = 36
12
b
Pythagorean Theorem
Substitute for a and c.
Simplify powers.
128 11.31
Try This: Example 3
Use the Pythagorean Theorem to find the height of the
triangle. Then use the height to find the area of the
triangle.
a2 + b2 = c2
a2 + 22 = 52
Pythagorean Theorem
Substitute for b and c.
1
(8)(
20) = 4 20 units2 17.89 units2
2
a2 + b2 = c2
42 + b2 = 122
16 + b2 = 144
–16
–16
b2 = 128
b 11.31
576 = 24
Find the square root of both sides.
Simplify powers.
Solve for c; c = c2.
Solve for the unknown side in the right triangle.
5
5
a
a2 = 20
a = 20 units ≈ 4.47 units
4
Pythagorean Theorem
Substitute for a and b.
Try This: Example 2
4
Additional Example 3: Using the Pythagorean Theorem to
Find Area
Use the Pythagorean Theorem to find the height of the
triangle. Then use the height to find the area of the
triangle.
a2 + b2 = c2
62 + 52 = c2
36 + 25 = c2
61 = c
7.81 c
2
A = 1hb =
2
Pythagorean Theorem
Substitute for b and c.
a2 + 4 = 25
a2 = 21
a = 21 units ≈ 4.58 units
2
Find the square root of both sides.
1
(4)(
21) = 2 21 units2 4.58 units2
2
7