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

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