The Pythagorean
Theorem
Section 8-1
 Use the Pythagorean Theorem.
Key Vocabulary
•
•
•
•
Leg
Hypotenuse
Pythagorean Theorem
Pythagorean Triple
Parts of a Right Triangle
• Longest side is the
hypotenuse, side c
(opposite the 90o
angle).
• The other two
sides are the legs,
sides a and b.
• Pythagoras
developed a
formula for finding
the length of the
sides of any right
triangle.
Theorem 4.7 - The Pythagorean
Theorem
The Pythagorean Theorem
In a right triangle, the square of
the length of the hypotenuse is
equal to the sum of the squares
of the lengths of the legs.
Example:
(hypotenuse)2=(leg)2+(leg)2
Example 1
Find the length of the hypotenuse.
SOLUTION
(hypotenuse)2 = (leg)2 + (leg)2
c2 = 52 + 122
c2 = 25 + 144
c2 = 169
c2 = 169
c = 13
ANSWER
Pythagorean Theorem
Substitute.
Multiply.
Add.
Find the positive square root.
Solve for c.
The length of the hypotenuse is 13.
Example 2
Find the unknown side length.
SOLUTION
(hypotenuse)2 = (leg)2 + (leg)2
142 = 72 + b2
196 = 49 + b2
196 – 49 = 49 + b2 – 49
147 = b2
147 = b2
12.1 ≈ b
ANSWER
Pythagorean Theorem
Substitute.
Multiply.
Subtract 49 from each side.
Simplify.
Find the positive square root.
Approximate with a calculator.
The side length is about 12.1.
Your Turn:
Find the unknown side length.
1.
ANSWER
8
ANSWER
8
ANSWER
about 10.6
2.
3.
Example 3a
A. Find x.
The side opposite the right angle is the hypotenuse,
so c = x.
a2 + b2 = c2
Pythagorean Theorem
42 + 72 = c2
a = 4 and b = 7
Example 3a
65 = c2
Simplify.
Take the positive square root
of each side.
Answer:
Example 3b
B. Find x.
The hypotenuse is 12, so c = 12.
a2 + b2 = c2
Pythagorean Theorem
x2 + 82 = 122
b = 8 and c = 12
Example 3b
x2 + 64 = 144
x2 = 80
Simplify.
Subtract 64 from each side.
Take the positive square
root of each side and
simplify.
Answer:
Your Turn:
A. Find x.
A.
B.
C.
D.
Your Turn:
B. Find x.
A.
B.
C.
D.
More Examples:
1)
2)
3)
4)
5)
6)
7)
8)
A=8, C =10 , Find B
A=15, C=17 , Find B
B =10, C=26 , Find A
A=15, B=20, Find C
A =12, C=16, Find B
B =5, C=10, Find A
A =6, B =8, Find C
A=11, C=21, Find B
B=6
B=8
A = 24
C = 25
B = 10.6
A = 8.7
C = 10
B = 17.9
C
A
B
Pythagorean Triples
Three whole numbers that work in the Pythagorean
formulas are called Pythagorean Triples.
The largest number in each triple is the length of the
hypotenuse.
Pythagorean triples are not the only possible side
lengths for a right triangle. They give the triangles
where all the lengths are whole numbers, but the side
lengths could be any real numbers.
Pythagorean Multiples
If you multiply the lengths of all three sides of
any right triangle by the same number,
then the resulting triangle is a right
triangle.
In other words, if a2 + b2 = c2, then
(an)2 + (bn)2 = (cn)2.
Therefore, additional pythagorean triples can
be found by multiplying each number in a
known triple by the same factor.
Pythagorean Triples
Multiples
Primitive Pythagorean Triples
A set of Pythagorean triples is considered a
primitive Pythagorean triple if the
numbers are relatively prime; that is, if
they have no common factors other than 1.
3-4-5
5-12-13
7-24-25
8-15-17
9-40-41
11-60-61
12-35-37
13-84-85
16-63-65
20-21-29
28-45-53
33-56-65
36-77-85
39-80-89
48-55-73
65-72-97
You need know the first 4 primitives: 3-4-5,
5-12-13, 7-24-25, 8-15-17.
Example 4
Use a Pythagorean triple to find x. Explain your
reasoning.
Example 4
Notice that 24 and 26 are multiples of 2 : 24 = 2 ● 12
and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean
triple, the missing leg length x is 2 ● 5 or 10.
Answer:
x = 10
Check:
242 + 102 =? 262
676 = 676 
Pythagorean Theorem
Simplify.
Your Turn:
Use a Pythagorean triple to find x.
A. 10
B. 15
C. 18
D. 24
More Practice
Use Pythagorean Triples to find each missing
side length.
Primitive: 5-12-13
X=26
Primitive: 7-24-25
X=50
Primitive: 3-4-5
X=15