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Sec 6.6 Pythagorean Theorem
Objective- To solve problems involving the Pythagorean Theorem.
For Right Triangles Only!
leg
leg
hypotenuse - always opposite tothe right angle
Objective- To solve problems involving the Pythagorean Theorem.
For Right Triangles Only!
Leg
leg
hypotenuse
Leg 2
Objective- To solve problems involving the Pythagorean Theorem.
For Right Triangles Only!
hypotenuseLeg 1
Leg 1
Leg 2
Hyp
Pythagorean Theorem
Objective- To solve problems involving the Pythagorean Theorem.
For Right Triangles Only!
(Leg1)2 + (leg2)2 = (Hyp)2
6
8
x
Solve for x.(Leg1)2 + (leg2)2 = (Hyp)2
€
62 + 82 = x 2
€
36 + 64 = x 2
€
100 = x 2
€
100 = x 2
€
x =10
€
x = −10
Line Segment can’t be negative.
4
7
y
Solve for y.(Leg1)2 + (leg2)2 = (Hyp)2
€
42 + 72 = y 2
€
16 + 49 = y 2
€
65 = y 2
€
65 = y 2
€
y = 65 ≈ 8.1
€
y = − 65 ≈ −8.1
Line Segment can’t be negative.
6
t15
Solve for t.(Leg1)2 + (leg2)2 = (Hyp)2
€
62 + t 2 =152
€
t 2 =189
€
t 2 = 189
€
t = 189 ≈13.7
Line Segment can’t be negative.
€
36 + t 2 = 225
Pythagorean Triples
3 4 5
6 8 10
9 12 15
12 16 20
Pythagorean Triples
3 4 5
6 8 10
9 12 15
12 16 20
5 12 13
10 24 26
15 36 39
7 24 25
14 48 50
21 72 75
915
12
To the nearest tenth of a foot, find the length of the diagonal of a rectangle with a width of 4 feet and a length of 10 feet.
4 ft.
10 ft.
x(Leg1)2 + (leg2)2 = (Hyp)2
€
42 +102 = x 2
€
16 +100 = x 2
€
116 = x 2
€
116 = x 2
€
116 = x
20 miles
A car drives 20 miles due east and then 45 milesdue south. To the nearest hundredth of a mile, how far is the car from its starting point?
45 milesx
2 2 2a b c 2 2 220 45 x
2400 2025 x 22425 x
22425 x
x 2425x 49.24
Application
• The Pythagorean theorem has far-reaching ramifications in
other fields (such as the arts), as well as practical applications.
• The theorem is invaluable when computing distances between
two points, such as in navigation and land surveying.
• Another important application is in the design of ramps. Ramp
designs for handicap-accessible sites and for skateboard parks
are very much in demand.
WallLadder
HYPLeg 1
Leg 2
Informal Proof #1Inscribe a square within the square.
a b
a
a
a
b
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b c
c
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c
2c
Informal Proof #12Yellow region c
a b
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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2c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #12Yellow region c
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Informal Proof #1
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2Yellow region c
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Informal Proof #1
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b c
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2Yellow region c
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Informal Proof #1
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2Yellow region c
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Informal Proof #1
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2Yellow region c
Informal Proof #1
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2b
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2Yellow region c
2 2Yellow region a b
2 2 2a b c
a b
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b
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b c
c
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2c
Informal Proof #2
a + bTotal Purple Yellow Area Area Area- =
2(a b) -1
4 ab2
=2c
2 2a 2ab b 2ab 2c
2 2 2a b c
