geometry 1 the pythagorean theorem. 2 a b c given any right triangle, a 2 + b 2 = c 2
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Geometry
1
The Pythagorean
Theorem
The Pythagorean Theorem
2
A
B
C
Given any right triangle, A2 + B2 = C2
Example3
A
B
C
In the following figure if A = 3 and B = 4,
Find C.
A2 + B2 = C2
32 + 42 = C 2
9 + 16 = C2
5 = C
Verifying the Pythagorean Theorem
4
Given a piece of graph paper, make a right triangle. Then make squares of the right triangle. Then find the square’s areas.
Pythagorean Theorem : Examples for finding the hypotenuse.
5
A=8, B= 15, Find C
A=7, B= 24, Find C
A=9, B= 40, Find C
A=10, B=24, Find C
A =6, B=8, Find C
A
B
C
C = 17
C = 25
C = 41
C = 26
C = 10
Finding the legs of a right triangle:
6
A
B
C
In the following figure if B = 5 and C = 13,
Find A.
A2 + B2 = C2
A2 +52 = 132
A2 + 25 = 169
A2 = 144
A = 12
More Examples:7
1) A=8, C =10 , Find B2) A=15, C=17 , Find B3) B =10, C=26 , Find A4) A=15, B=20, Find C5) A =12, C=16, Find B6) B =5, C=10, Find A7) A =6, B =8, Find C8) A=11, C=21, Find B
A
B
C
B = 6
B = 8A = 24
C = 25B = 10.6A = 8.7C = 10B = 17.9
Given the lengths of three sides, how do you know if you have a right triangle?
8
A
B
C
Given A = 6, B=8, and C=10, describe the triangle.
A2 + B2 = C2
62 +82 = 102
36 + 64 = 100
* This is true, so you have a right triangle.
Pythagorean Triples
9
Some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles.
The three whole number side-lengths are called a Pythagorean triple.
•The 3-4-5 triangle • An example is a = 3, b = 4 and h = 5, called "the 3-4-5 triangle". We can check it as follows:
Pythagorean TriplesNot only is 3-4-5 a Pythagorean triple, but
so is any multiple of 3-4-5. 3-4-5 6-8-10 12-16-20 15-20-25 18-24-30
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Can you think of any others??
5,12,13 7,24,2520,21,29
What happens if it the Pythagorean Theorem does NOT work? If you do not have a picture nor an
angle that you know for a fact is 90 degrees, then it is possible to have an acute or an obtuse triangle.
If A2 + B2 > C2, you have an acute triangle.
If A2 + B2 < C2, you have an obtuse triangle.
11
If A2 + B2 > C2, it is an acute triangle.
12
Given A = 4, B = 5, and C = 6, describe the triangle.
A2 + B2 = C2
42 + 52 = 62
16 + 25 = 36
41 > 36, so we have an acute triangle.
A B
C
If A2 + B2 < C2, it is an obtuse triangle.
13
Given A = 4, B = 6, and C =8, describe the triangle.
A2 + B2 = C2
42 + 62 = 82
16 + 36 = 64
52 < 64, so we have an obtuse triangle.
A
C
B
Describe the following triangles asacute, right, or obtuse
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1) A=9, B=40, C=412) A=10, B=15, C=203) A=2, B=5, C=64) A=12, B=16, C=205) A=11, B=12, C=146) A=2, B=3, C=47) A=1, B=7, C=78) A=90, B=120,
C=150
A
B
C
right
acute
obtuse
right
right
right
obtuse
acute
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