unit 6 lesson 3 the pythagorean converse ccss g-srt 4: use congruence and similarity criteria for...
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Unit 6 Lesson 3 The Pythagorean Converse
CCSS
G-SRT 4: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Lesson Goals► Apply the Pythagorean
Converse to classify a triangle according to angle measure.
ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
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2
Draw
• In the Column Q’s section of your notes, attempt to draw a triangle with 1,1,and 3 cm sides.
• Then attempt to draw one with 4,4, and 3 cm sides
• Then attempt to draw one with 3,3, and 5 cm sides.
• Analyze and compare their differences with a student next to you.
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Theorem
Triangle Inequality TheoremThe length of the longest side of a triangle must be less than the sum of the lengths of the two shorter sides. A
BC
BC AB AC
B C
CB A
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Can a triangle be constructed with sides of the following measures?
5, 7, 8
The length of the longest of a triangle must be less
than the sum of the lengths of the two shorter sides.
8 < 5 + 7
You Try
Yes
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Can a triangle be constructed with sides of the following measures?
4.2, 4.2, 8.4
The length of the longest of a triangle must be less
than the sum of the lengths of the two shorter sides.
8.4 < 4.2 + 4.2
You Try
NO
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Can a triangle be constructed with sides of the following measures?
3, 6, 10
10 < 3 + 6
You Try
NO
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Can a triangle be constructed with sides of the following measures?
3, 3, 8
8 < 3 + 3
You Try
NO
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Can a triangle be constructed with sides of the following measures?
9, 5, 11
11 < 9 + 5
You try
Yes
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2 2 2If then it is an obtuse triangle.c a b
2 2 2If then it is an acute triangle.c a b
2 2 2If then it is a right triangle.c a b
Theorem
The Pythagorean Converse
Keep the longest length on the left!
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Classify the triangle as right, acute, or obtuse.
example
78
113
113 113
113 ? 64 49
22 2113 ? 8 7
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7
10
13
169 149
169 ? 100 49
2 2 213 ? 10 7
exampleClassify the triangle as right, acute, or obtuse.
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exampleDecide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute, or obtuse.
8, 18, and 24
?
24 8 18
24 26
To be a triangle, the longest side must be less than the sum of other two sides.
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example
8, 18, and 24
Use the Pythagorean Converse to classify the triangle.2 2 224 ? 8 18
576 ? 64 324
576 388
Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute, or obtuse.
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You TryDecide whether the set of numbers can represent the side lengths of a triangle.
32, 48, and 51
To be a triangle, the longest side must be less than the sum of other two sides.
?
51 32 48
51 80
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You TryClassify the triangle as right, acute, or obtuse.
32, 48, and 51
Use the Pythagorean Converse to classify the triangle.
2 2 251 ? 32 48
2601 ? 1024 2304
2601 3328
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You TryDecide whether the set of numbers can represent the side lengths of a triangle.
8, 40, 41?
41 8 40
41 48
2 2 241 ? 8 40
1681 ? 64 1600
1681 1664
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You TryDecide whether the set of numbers can represent the side lengths of a triangle.
12.3, 16.4, 20.5?
20.5 12.3 16.4
20.5 28.7
2 2 220.5 ? 12.3 16.4
420.25 ? 151.29 289.96
420.25 420.25
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2 65c
260c
2 260c
2 64 196c
2 2 28 14c
Find the range of values for c, the longest side of the triangle, so that the triangle is acute when a = 8 and b = 14.
8c
14A
B
C
14 16.1c 16.1
example
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3 41c
369c
2 369c
2 144 225c
2 2 212 15c
Find the range of values for c, the longest side of the triangle, so that the triangle is obtuse when a = 12 and b = 15.
12c
15A
B
C
19.2 27c 19.2
example
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369c
2 305c
2 49 256c
2 2 27 16c
7c
16A
B
C
17.5 23c
17.5
You TryFind the range of values for c, the longest side of the triangle, so that the triangle is obtuse when a = 7 and b = 16.
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2 21 5 7 5
Example
B
A
C
AB 2 25 1 2 7 2 24 5 41
BC 2 26 2 40
AC 2 25 5 2 5 2 210 3 109
2 2 2
109 ? 40 41
109 ? 40 41
109 81
Obtuse Triangle
Graph points 5,2 , 1,7 , and 5,5 . Connect the points to form .Decide whether is right, acute, or obtuse.
A B CABC
ABC
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Summary
• Create an acronym, poem, or mnemonic to help you remember the Pythagorean Converse.
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Today’s Assignment
p. 546: 14 – 20 e; 32, 33, 34 +
Find the value for c, the longest side of the triangle, so that the triangle is a) acute and b) obtuse.
a) 5, 11
b) 12, 17
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Find the value for c, the longest side of the triangle, so that the triangle is a) acute and b) obtuse.
+1) 5, 11 +2) 12, 17