Geometry

Lesson 6.4Inequalities

for One Triangle

Tomorrow we will learn inequalitiesFor Two Triangles

The Main Idea

You can think of an angle as the jaw bones of an alligator.

When you prop the jaw open with a stick, you form a triangle.

The larger the stick, the larger the angle formed by the jawbones.

The smaller the stick, the smaller the angle formed by the jawbones.

You could prop a very long stick in the mouth of a hippo, creating a very large angle.

long stick

largeangle

It follows logically that the longest side of a triangle would be opposite the largest angle

And the shortest side would be opposite the

smallest angle.

Theorem

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

A

C B

If BC > AB, then

big

smallm<A > m<C.

                                                                                                           

   

                                                        

    

                                                                                                                               

Crikey,that seems easy! Longest side opposite biggest angle, shortest side opposite smallest angle, medium side opposite medium angle.

A

B C

10

6

9

A

B C

x -3x + 10

x

Examples: What ’s are the largest and smallest?

1.

Largest : _____; Smallest : ______

2.

Largest : _____; Smallest : ______

A

B C3060

90A

B C

5x

2x 10x

Examples: What sides are the largest and smallest?

3.

Largest side: _____; Smallest side:_____

4.

Largest side: _____; Smallest side:_____

Consider two sides of a triangle, each side 6 inches

long

What is the maximum length the third side can be?

6 6

3

5

9

6 6

3 5 9

6 66 6

6 6

11.999

How long could the third side be?

Could it be 12 or more inches long?

6 6

12 No – ∆ could NOT be drawn

Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

A

C B11

914

9 + 11 = 20 20 > 14

11 + 14 = 25 25 > 9

14 + 9 = 23 23 > 11

ExampleThe lengths of two sides of a triangle are 3 and 5. The length of the third side must be greater than ________ and less than _______.

Let x be the length of the third side.

x + 3 > 5

x >

3 + 5 > x

8 > x

x <

x + 5 > 3

x > -228

Examples: Is this a triangle with these side measures possible?

5. 10, 9, 8 6. 16, 11, 5

B C

A

x + 3x

x - 3

A

x - 5x

B C

A

B C

D

5254

53

58

60

59

60A B

CD

A

B

C

D

68

3x

11x

4x

A

54B C

Sample Problems

3. ABCD is parallelogram

Largest : ____ _____;

1.

Largest : ____; Smallest : ____

2.

Largest : ____; Smallest : ____ Smallest : ________

4.

Largest side: _________Smallest side: _______

5.

Largest side: ________Smallest side: ________

6.

Longest Side: ____

Given the possible sides listed, do we have a triangle:

7. 5, 11, 15

8. 10, 10, 0.5

The 3rd side of this triangle must be between what two numbers:

9. Sides: 5, 15

_____ < 3rd side <______

10. Sides: 20, 14

_____ < 3rd side <_____

The 3rd side of this triangle must be between what two numbers:

11. Sides: 4, 5

_____ < 3rd side <______

12. Sides: 3x, 7x

_____ < 3rd side <______

Homework

pg. 222 #1-17

Corollaries

The segment from a point to a line is the shortest segment from the point to the line.

The segment from a point to a plane is the shortest segment from the point to the plane.

●

●

The following is an Anti- Drug Message…….

Examples

6 10

8B

A

C

Which is the largest angle?<B

Which is the smallest angle?<C

20˚40˚

120˚

A

B C

Name the side lengths from shortest to longest.

BCABAC

Nude crack man in alligator rescue

Thursday, November 30, 2006

In further news of the misadventures of people who are naked, a naked man on crack was pulled from the jaws of an alligator in Florida on Wednesday.

He was only saved when sheriff's deputies jumped into the lake where he was being eaten, and wrestled with the alligator until it let go of the man, before pulling the man about 40m back to the shore.

They had responded to several reports from members of the public that screams had been heard coming from the lake, at around 4 o'clock in the morning.

The man – identified as 45-year-old Adrian Apgar – lost his left arm in the alligator attack, as well as suffering severe injuries to his right arm and left leg. His right leg is believed to be fine.

'He was totally naked,' noted Sheriff Grady Judd.

He admitted to the deputies after he was pulled from the water that he'd been smoking crack. No explanation has yet been offered for why he was naked, although police have not ruled out the possibility that he might have decided to go for a swim.

Judd added: 'He admitted that he'd been smoking crack cocaine. But still, it's a human life. Our deputies don't ask questions, they respond and they save people.'

At the time of writing, the condition of the alligator is unknown.

Getty Images

In MemorySteve Irwin1962 - 2006