Geometry 9.3 Converse of the Pythagorean Theorem.

Geometry

9.3 Converse of the Pythagorean Theorem

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 2

Goals

I can determine if a triangle is a right triangle.

I can use the Pythagorean inequalities to determine if a triangle is acute or obtuse.

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 3

Pythagorean Theorem In 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.

If ABC is a right triangle, then a2 + b2 = c2

a

b

c

A

B

C

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 4

Converse of Pythagorean Theorem

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

If a2 + b2 = c2, then ABC is a right triangle.

a

b

c

A

B

C

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 5

Example 1

Is POD a right triangle?

P

O

D

30 16

34

?2 2 2

?

16 30 34

256 900 1156

1156 1156

Yes!Longest Side

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 6

Reminder

22

2

2

5 5

17 17

3 3

x x

x x

2 2 2 2

2 22

2 22

2

3 3 9

3 3 9

3 3 3 3 9 3 27

4 5 16 5 80

x x x

x x x

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 7

Example 2

Is HUG a right triangle?

5 5

H

U G

5

10

Which segment is the longest? HG

? 22 2

? 22

?

5 10 5 5

25 100 5 5

125 25 5

125 125

Yes!

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 8

Example 3

Is SAD a right triangle?

S

A D

9

12

Which segment is the longest? SD

?2 2 2

?

9 12 20

81 144 400

225 400

No!

20

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 9

Your Turn.

Is RST a right ?

R

S

T26

1024

?2 2 2

?

10 24 26

100 576 676

676 676

Yes it is.

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 10

Triangle Inequality Theorem

In a triangle, the sum of any two sides is greater than the third side.

4

5

74 + 7 > 5

4 + 5 > 7

5 + 7 > 4

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 11

Triangle Inequality Theorem

5

10

4This is not a triangle since 5 + 4 < 10.

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 12

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 13

a

b

c

Begin with a right triangle…

a2 + b2 = c2

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 14

a

b

ca

c

a and b have not changed.

a2 + b2 has not changed.

c got smaller.

c2 got smaller.

and…

The right angle gets smaller: it is acute.

Rotate side a in.

c2 = a2 + b2c2 < a2 + b2

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 15

Theorem 9.6

If the square of the length of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute.

A

BC a

bc

c2 < a2 + b2

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 16

a

b

c

Take another right triangle…

a2 + b2 = c2

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 17

a

b

ca c

a and b have not changed.

a2 + b2 has not changed.

c got larger.

c2 got larger.

and…

The right angle gets larger: it is obtuse.

Rotate side a out.

c2 = a2 + b2c2 > a2 + b2

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 18

Theorem 9.6 If the square of the length of the

longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse.

c2 > a2 + b2

A

BC a

bc

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 19

Example 4

The sides of a triangle measure 5, 7, and 11. Classify it as acute, right, or obtuse.

Solution: The longest side is 11. 112 ? 52 + 72

121 ? 25 + 49 121 > 74 Obtuse

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Example 5

The sides of a triangle are 17, 20, and 25. Classify the triangle.

Solution: 252 ? 172 + 202

625 ? 689 625 < 689 Acute

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Example 6

Classify this triangle.

57

12

2 2 212 ____ 7 5

12____ 7 5

12 12

?

?

Right

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Example 7

Classify this triangle.

It isn’t a triangle! 6 +8 < 16.

68

16

April 21, 2023 Geometry 9.3 Converse of the Pythagorean Theorem 23

Summary

If c2 = a2 + b2, RIGHT . If c2 < a2 + b2, ACUTE . If c2 > a2 + b2, OBTUSE . The last two can be very confusing;

don’t get them mixed up.