5-3a the pythagorean theorem
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5-3A The Pythagorean Theorem. You used the Pythagorean Theorem to develop the Distance Formula. Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem. c. a. b. Pythagorean Theorem. - PowerPoint PPT PresentationTRANSCRIPT
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5-3A The Pythagorean Theorem
You used the Pythagorean Theorem to develop the Distance Formula.
• Use the Pythagorean Theorem.
• Use the Converse of the Pythagorean Theorem.
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Pythagorean TheoremThe Pythagorean Theorem is used to calculate
the length of any side of a right triangle when the lengths of the other two sides are known.
Which side is the hypotenuse?
Which sides are the legs?a
b
c
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Pythagorean TheoremIn 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.
a2 + b2 = c2
b
ac
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Find the length of the third side of the right triangle ∆STU.
ST=3, TU = 4
ST2 + TU2 = SU2
32 + 42 = SU2
9 + 16 = SU2
25 = SU2
5 = SU
S
T U
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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 = 765 = c2 Simplify.
Take the positive square rootof each side.
Answer:
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B. Find x.
The hypotenuse is 12, so c = 12.
a2 + b2 = c2 Pythagorean Theorem
x2 + 82 = 122 b = 8 and c = 12
Take the positive squareroot of each side andsimplify.
x2 + 64 = 144 Simplify.
x2 = 80 Subtract 64 from each side.
Answer:
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Find the length of the third side of the right triangle ∆STU.
ST=7, SU = 10ST2 + TU2 = SU2
72 + TU2 = 102
49 + TU2 = 102
49 + TU2 = 100TU2 = 51
14.751TU
S
T U
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A. Find x.
A.
B.
C.
D.
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B. Find x.
A.
B.
C.
D.
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Pythagorean Triple Pythagorean Triples are special sets of
numbers that all the numbers are positive integers.
A Pythagorean triple is 3, 4, and 5.
32 + 42 = 52
9 + 16 = 25
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Pythagorean Triples under 100
(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)
Formula:
Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple.
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Find the length of the missing side.
8
610
Triple = 6, 8, 10
25 7
24
Triple = 7, 24, 25
12
13
5
Triple = 5, 12, 13
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Use a Pythagorean triple to find x. Explain your reasoning.
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 Pythagorean Theorem?
676 = 676 Simplify.
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A. 10
B. 15
C. 18
D. 24
Use a Pythagorean triple to find x.
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If you have the lengths of three sides of a triangle, you can use the converse of the Pythagorean Theorem to prove it is a right triangle.
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You can also use side lengths to classify an acute or obtuse triangle.
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A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.
9 + 12 > 15 9 + 15 > 12 12 + 15 > 9
The side lengths 9, 12, and 15 can form a triangle.Step 2 Classify the triangle by comparing the
square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2.
?
152 = 122 + 92 Substitution?
225 = 225 Simplify and compare.Answer: Since c2 = a2 + b2, the triangle is a right triangle.
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B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem.
10 + 11 > 13 10 + 13 > 11 11 + 13 > 10
The side lengths 10, 11, and 13 can form a triangle.
Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides.
c2 = a2 + b2 Compare c2 and a2 + b2.
?
132 = 112 + 102 Substitution?
169 < 221 Simplify and compare.
Answer: Since c2 < a2 + b2, the triangle is acute.
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A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.
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• What is the Pythagorean Theorem?
a2 + b2 = c2
• Why is it important?
It is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known.
• What is a Pythagorean Triple?
Pythagorean Triples are special sets of numbers that all the numbers are positive integers.
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8-2 Assignment
Page 552, 8-28 even