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
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.
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
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
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
Can a triangle be constructed with sides of the following measures?
3, 6, 10
10 < 3 + 6
You Try
NO
Can a triangle be constructed with sides of the following measures?
3, 3, 8
8 < 3 + 3
You Try
NO
Can a triangle be constructed with sides of the following measures?
9, 5, 11
11 < 9 + 5
You try
Yes
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!
Classify the triangle as right, acute, or obtuse.
example
78
113
113 113
113 ? 64 49
22 2113 ? 8 7
7
10
13
169 149
169 ? 100 49
2 2 213 ? 10 7
exampleClassify the triangle as right, acute, or obtuse.
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.
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.
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
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
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
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
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
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
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.
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
Summary
• Create an acronym, poem, or mnemonic to help you remember the Pythagorean Converse.
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
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