Download The Pythagorean Theorem

The Pythagorean Theorem
Geometry
Mrs. Pam Miller
January 2010
The Pythagorean Theorem
• Greek Mathematician, Pythagoras, proved this theorem.
• Applies to right triangles.
• Many different proofs exist, including one by President
Garfield.
The Theorem
For any right triangle, the square of the
hypotenuse is equal to the sum of the
squares of the legs.
hypotenuse
leg
leg2 + leg2 = hyp2
leg
The Abbreviated Version
We often see the Pythagorean
Theorem stated as:
a 2 + b2 = c2
c
a
b
Some Examples
Find the value of x.
( Remember that the length of a segment must be a positive number.)
leg2 + leg2 = hyp2
a)
x
3
32 + 72 = x2
9 + 49 = x2
7
58 = x2
√58 = √x2
√58 = x
Examples (cont’d.)
8
b)
10
leg2 + leg2 = hyp2
x
82 + x2 = 102
64 + x2 = 100
64 - 64 + x2 = 100 - 64
X2 = 36
√x2 = √36
X=6
Practice with Radicals
Work with the Pythagorean Theorem often requires us to work with radicals.
Simplify each expression:
A) (√3) 2
√3 × √3
√9
3
B) ( 3 √ 11 ) 2
3 √ 11 × 3 √ 11
9 × √ 121
9
99
× 11
Your Turn
Simplify each expression:
A. (√ 5) 2
B. (2 √ 7) 2
E.
F.
2
 3 


 5
 2
 
 2 
 
2
C. (7 √ 2 ) 2
G.
D. (2n) 2
2

3

6

2
The Answers
Simplify each expression:
A. (√ 5) 2
=5
B. (2 √ 7) 2
= 28
C. (7 √ 2 ) 2
= 98
E.
F.
G.
D. (2n) 2
= 4n2
2
 3 


 5
 2
 
 2 
 
= 9/5
2
= 1/2
2
 2  = 24/9
 6
3 
Pythagorean Triples
3, 4, 5
5, 12, 13
6, 8, 10
10, 24, 26
8, 15, 17
7, 24, 25
9, 12, 15
12, 16, 20
15, 20, 25
In every triple, the largest # is
the length of the hypotenuse
and the 2 smaller numbers are
the lengths of the legs of the
right triangle.
The Converse
If the square of 1 side of a triangle is equal to the sum of the squares of the
other two sides, then the triangle is a right triangle.
More simply…
if c2 = a2 + b2,
then the triangle is a
right triangle.
c
a
b
Pythagorean Inequalities
If c2 > a2 + b2,
then the triangle is obtuse.
If c2 < a2 + b2,
then the triangle is acute.
c
a
b
Practice
The sides for 3 triangles are given.
Decide if each triangle is acute, right or obtuse.
A) 14, 7, 9
B) 2.5, 6, 6.5
C) 2, 3, 3.5
142 ___ 72 + 92
6.52 ___ 2.52 + 62
3.52 ___ 22 + 32
196 ___ 49 + 81
42.25 ___ 6.25 + 36
12.25 ___ 4 + 9
196 > 130
42.25 = 42.25
12.25 < 13
It’s obtuse!
It’s right!
It’s acute!
Check It Out
A triangle has sides with the following lengths: 9, 40, & 41. Is this a
right triangle?
Does 412 = 92 + 402 ?
Does 1681 = 81 + 1600?
YES !! So, the triangle is a right triangle.
More Practice
A right triangle has one leg with a length of 48 and a hypotenuse with a length
of 80.
What is the length of the other leg?
Here’s how:
482 + x2 = 802
x2 = 4096
2304 + x2 = 6400
√ x2 = √ 4096
x2 = 6400 - 2304
x = 64
64
Practice (cont’d)
A triangle has side lengths of 7, 10, & 12.
Is the triangle a right triangle?
Use the Converse of the Pythagorean Theorem!
Here’s How:
Does c2 = a2 + b2 ?
Does 122 = 72 + 102 ?
Does 144 = 49 + 100?
NO !
Practice (cont’d)
A triangle has side lengths of 8, 15, and 18.
Is the triangle right, acute, or obtuse?
Remember:
If c2 < a2 + b2, you have an acute triangle.
If c2 = a2 + b2, you have a right triangle.
If c2 > a2 + b2, you have an obtuse triangle.
Here’s How:
182 ____ 82 + 152
324 > 289
324 ____ 64 + 225
It’s obtuse!
Other Applications
Find the area of the figure.
Leave your answer in radical form.
8
8
Area of a triangle = 1/2 bh
8
Use the Pythagorean Theorem to find “h”:
h
8
4
42 + h2 = 82
h2 = 48
16 + h2 = 64
√ h2 =
h2 = 64 - 16
h=4√ 3
√ 48
More….
Find the Area of the Triangle
Area = 1/2 bh
8
4√ 3
4
Area = 1/2 (8) (4 √ 3)
Area = 16 √ 3
8
8
8
Last Problem!
Find the area of the square. Leave your answer in radical form.
Use the Pythagorean Theorem to find the length of the
sides (s).
s2 + s2 = 62
√ s2 = √ 18
2s2 = 36
s=3√ 2
s2 = 18
So the area of the square is :
3√ 2×3√ 2
6
s
s
=9√ 4
= 9×2
= 18
What Have You Learned?
•
•
•
•
•
Pythagorean Theorem
Converse of the Pythagorean Theorem
Pythagorean Triples
Pythagorean Inequalities
Applications