Geometry 9.2 The Pythagorean Theorem
Jan 12, 2016
Geometry
9.2 The Pythagorean Theorem
April 21, 2023 Geometry 9.2 The Pythagorean Theorem 2
Goals
Prove the Pythagorean Theorem. Solve triangles using the theorem. Solve problems using the theorem.
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This is ancient history.
The Egyptian Pyramid builders used it to make square corners.
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Terminology
Leg
Leg
Hypotenuse
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Hypotenuse = “stretched against”
35
4
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Proof
Proofs of the Pythagorean Theorem are numerous – well over 300 known.
Discovered in many ancient cultures. We can use what we learned about the
altitude of right triangles to prove the Pythagorean Theorem, too.
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From last lesson:
2 2
a cm b cn
a cm b cn
D
ab
mn
h
c
2 2
2 2 2
( )
a b cm cn
c m n
c c
a b c
April 21, 2023 Geometry 9.2 The Pythagorean Theorem 8
Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
April 21, 2023 Geometry 9.2 The Pythagorean Theorem 9
Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
Area of the square:
A = c2
Area of one triangle:
A = (½) ab
Area of 4 triangles:
A = 2ab
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Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
Area of the square:
A = c2
Area of 4 triangles:
A = 2ab
Area Sum
c2 + 2ab
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Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
Area Sum
c2 + 2ab
?a + b
?a + b
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Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
Area Sum
c2 + 2ab
Area another way:
2
2 2
2 2
( )
( )( )
2
A a b
a b a b
a ab ab b
a ab b
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Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
Area Sum
c2 + 2ab
or
a2 + 2ab + b2
These areas are equal.
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Chinese Proof
a b
a
a
a
b
b
b
c
c
c
c
2 2 2
2 2 2
2 2a ab b c ab
a b c
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President Garfield (1876)20th President of the United States
Area of Trapezoid = Sum of area of three triangles
21 1 1 12 2 2 2
2 2 21 12 2
2 2 2
2 2 2
2
2 2
a b a b ab ab c
a ab b ab c
a ab b ab c
a b c
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The Pythagorean Theorem
a
b
c
a2 + b2 = c2
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Example 1 Solve.
5
6
c
2 2 25 6
25 36
61
61
7.81
c
c
c
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Example 2 Solve.
a
2
10
2 2 2
2
2
2 10
4 100
96
96 16 6
4 6 9.80
a
a
a
a
a
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Example 3 Solve.
x
x20
2 2 2
2
2
20
2 400
200
200
10 2
14.14
x x
x
x
x
x
x
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Solve these two triangles.
2 2 2
2
2
3 4
9 16
25
5
c
c
c
c
3
4
c5
12
c
2 2 2
2
2
5 12
25 144
169
13
c
c
c
c
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Pythagorean Triples
3
4
55
12
13
3 – 4 – 5 and 5 – 12 – 13 are Pythagorean Triples.
Each side is an integer.
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Example
Is 10-10-20 a Pythagorean Triple? 102 + 102 = 202 ? 100 + 100 = 400 ? 200 = 400 ? False! Not a Pythagorean Triple.
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Example
Is 20-21-29 a Pythagorean Triple? 202 + 212 = 292 ? 400 + 441 = 841 ? 841 = 841 True It is a Pythagorean Triple.
Generating Triples
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Area of a Triangle
h
b
12A bh
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Find the area.
12
15
12A bh
h
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Find the area.
12
15h
2 2 2
2
2
12 15
144 225
81
9
h
h
h
h
A Pythagorean Triple
9
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Find the area.
12
159
12 12 9
6(9)
54
A
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Problem
The distance between bases on a baseball diamond is 90 feet. A catcher throws the ball from home base to 2nd base. What is the distance?
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Problem
90
90
2 2 2
2
2
90 90
8100 8100
16200
16200
127.3
c
c
c
c
c
c
April 21, 2023 Geometry 9.2 The Pythagorean Theorem 30
Find the diagonal measure of the LCD screen to the nearest inch.
36.8 in.
20.7 in.
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Find the diagonal measure of the LCD screen to the nearest inch.
36.8 in.
20.7 in.c
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Find the diagonal measure of the LCD screen to the nearest inch.
2 2 2
2
2
36.8 20.7
1354.24 428.49
1782.73
42.22
c
c
c
c
36.8 in.
20.7 in.c
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Find the diagonal measure of the LCD screen to the nearest inch.
2 2 2
2
2
36.8 20.7
1354.24 428.49
1782.73
42.22
c
c
c
c
36.8 in.
20.7 in.42 in.
About 42 inches
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Summary
In a right triangle, the hypotenuse is the longest side.
The sum of the squares of the legs is equal to the square of the hypotenuse.
If the three sides are all integers, they form a Pythagorean Triple.
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True or False? a b
c
a + b = c?http://www.youtube.com/watch?v=DUCZXn9RZ9s
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False. It should have been…
The sum of the squares of the two legs of a
right triangle is equal to the
square of the remaining side. Oh joy! Rapture! I have a brain!
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HomeworkHow to Generate
Pythagorean Triples
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Generating Pythagorean Triples
Find two positive integers a & b which are relatively prime and a > b. That is, they have no factors in common other than 1.
Then the triples are: a2 + b2 , 2ab and a2 – b2.
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Generating Pythagorean Triples
Example: Choose a = 4 and b = 3. a2 + b2 = 42 + 32 = 25. 2ab = 2(4)(3) = 24. a2 – b2 = 42 – 32 = 7. 7, 24, 25 is a Pythagorean Triple.
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Generating Pythagorean Triples
7, 24, 25 is a Pythagorean Triple. Check: 72 + 242 = 252 ? 49 + 576 = 625 ? 625 = 625 That’s a triple!
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Pythagorean Triples
a and b are relatively prime. a > b a2 + b2
2ab a2 – b2
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Try it.
Using a = 8 and b = 3, find the Pythagorean Triple.
Answer: 82 – 32 = 64 – 9 = 55 2(8)(3) = 48 82 + 32 = 73 552 + 482 = 732? 3025 + 2304 = 5329 ? 5329 = 5329 checks.
Area