Loading...

PPT – Sec Math II PowerPoint presentation | free to download - id: 6f2faa-ZmU0Y

The Adobe Flash plugin is needed to view this content

Similarity of Triangles

- Sec Math II
- Unit 8 Lesson 3
- Class Notes

Click one of the buttons below or press the

enter key

EXIT

BACK

NEXT

In geometry, two polygons are similar when one is

a replica (scale model) of the other.

EXIT

BACK

NEXT

Consider Dr. Evil and Mini Me from Mike Meyers

hit movie Austin Powers. Mini Me is supposed to

be an exact replica of Dr. Evil.

EXIT

BACK

NEXT

EXIT

BACK

NEXT

The following are similar figures.

I

II

EXIT

BACK

NEXT

The following are non-similar figures.

I

II

EXIT

BACK

NEXT

Feefee the mother cat, lost her daughters, would

you please help her to find her daughters. Her

daughters have the similar footprint with their

mother.

Feefees footprint

EXIT

BACK

NEXT

1.

Which of the following is similar to the above

triangle?

B

A

C

EXIT

BACK

NEXT

Similar triangles are triangles that have the

same shape but not necessarily the same size.

?ABC ? ?DEF

When we say that triangles are similar there are

several repercussions that come from it.

?A ? ?D

?B ? ?E

?C ? ?F

Six of those statements are true as a result of

the similarity of the two triangles. However, if

we need to prove that a pair of triangles are

similar how many of those statements do we need?

Because we are working with triangles and the

measure of the angles and sides are dependent on

each other. We do not need all six. There are

three special combinations that we can use to

prove similarity of triangles.

1. PPP Similarity Theorem ? 3 pairs of

proportional sides

2. PAP Similarity Theorem ? 2 pairs of

proportional sides and congruent angles between

them

3. AA Similarity Theorem ? 2 pairs of

congruent angles

1. PPP Similarity Theorem ? 3 pairs of

proportional sides

?ABC ? ?DFE

2. PAP Similarity Theorem ? 2 pairs of

proportional sides and congruent angles between

them

m?H m?K

?GHI ? ?LKJ

The PAP Similarity Theorem does not work

unless the congruent angles fall between the

proportional sides. For example, if we have the

situation that is shown in the diagram below, we

cannot state that the triangles are similar. We

do not have the information that we need.

Angles I and J do not fall in between sides GH

and HI and sides LK and KJ respectively.

3. AA Similarity Theorem ? 2 pairs of

congruent angles

m?N m?R

?MNO ? ?QRP

m?O m?P

It is possible for two triangles to be similar

when they have 2 pairs of angles given but only

one of those given pairs are congruent.

m?T m?X

m?S m?Z

m?S 180?- (34? 87?)

?TSU ? ?XZY

m?S 180?- 121?

m?S 59?

Note One triangle is a scale model of the other

triangle.

EXIT

BACK

NEXT

How do we know if two triangles are similar or

proportional?

EXIT

BACK

NEXT

Triangles are similar () if corresponding angles

are equal and the ratios of the lengths of

corresponding sides are equal.

EXIT

BACK

NEXT

Interior Angles of Triangles

The sum of the measure of the angles of a

triangle is 1800. Ð A Ð B ÐC 1800

EXIT

BACK

NEXT

Example 6-1b

Determine whether the pair of triangles is

similar. Justify your answer.

Answer Since the corresponding angles have

equal measures, the triangles are similar.

If the product of the extremes equals the product

of the means then a proportion exists.

EXIT

BACK

NEXT

This tells us that ? ABC and ? XYZ are similar

and proportional.

EXIT

BACK

NEXT

Q Can these triangles be similar?

EXIT

BACK

NEXT

AnswerYes, right triangles can also be similar

but use the criteria.

EXIT

BACK

NEXT

EXIT

BACK

NEXT

Do we have equality?

This tells us our triangles are not similar. You

cant have two different scaling factors!

EXIT

BACK

NEXT

If we are given that two triangles are similar or

proportional what can we determine about the

triangles?

EXIT

BACK

NEXT

The two triangles below are known to be similar,

determine the missing value X.

EXIT

BACK

NEXT

EXIT

BACK

NEXT

In the figure, the two triangles are similar.

What are c and d ?

EXIT

BACK

NEXT

In the figure, the two triangles are similar.

What are c and d ?

EXIT

BACK

NEXT

EXIT

BACK

NEXT

Two triangles are called similar if their

corresponding angles have the same measure.

?

?

?

?

?

?

Two triangles are called similar if their

corresponding angles have the same measure.

?

Ratios of corresponding sides are equal.

C

A

?

a

c

?

?

?

?

b

B

a

A

b

B

c

C

Mary is 5 ft 6 inches tall. She casts a 2 foot

shadow. The tree casts a 7 foot shadow. How tall

is the tree?

Mary is 5 ft 6 inches tall. She casts a 2 foot

shadow. The tree casts a 7 foot shadow. How tall

is the tree?

Marys height

Trees height

Marys shadow

Trees shadow

x

5.5

2

7

Mary is 5 ft 6 inches tall. She casts a 2 foot

shadow. The tree casts a 7 foot shadow. How tall

is the tree?

5.5

x

2

7

Marys height

Trees height

Marys shadow

Trees shadow

x

5.5

2

7

5.5

x

2

7

7 ( 5.5 ) 2 x 38.5 2 x

x 19.25

The height of the tree is 19.25 feet

Congruent Figures

- In order to be congruent, two figures must be the

same size and same shape.

Similar Figures

- Similar figures must be the same shape, but their

sizes may be different.

Similar Figures

- This is the symbol that means similar.
- These figures are the same shape but different

sizes.

SIZES

- Although the size of the two shapes can be

different, the sizes of the two shapes must

differ by a factor.

4

2

6

6

3

3

2

1

SIZES

- In this case, the factor is x 2.

4

2

6

6

3

3

2

1

SIZES

- Or you can think of the factor as 2.

4

2

6

6

3

3

2

1

Enlargements

- When you have a photograph enlarged, you make a

similar photograph.

X 3

Reductions

- A photograph can also be shrunk to produce a

slide.

4

Determine the length of the unknown side.

15

12

?

4

3

9

These triangles differ by a factor of 3.

15 3 5

15

12

?

4

3

9

Determine the length of the unknown side.

?

2

24

4

These dodecagons differ by a factor of 6.

?

2 x 6 12

2

24

4

Sometimes the factor between 2 figures is not

obvious and some calculations are necessary.

15

12

10

8

18

12

?

To find this missing factor, divide 18 by 12.

15

12

10

8

18

12

?

18 divided by 12 1.5

The value of the missing factor is 1.5.

15

12

10

8

18

12

1.5

When changing the size of a figure, will the

angles of the figure also change?

?

40

?

?

70

70

Nope! Remember, the sum of all 3 angles in a

triangle MUST add to 180 degrees. If the size of

the angles were increased, the sum would

exceed 180 degrees.

40

40

70

70

70

70

We can verify this fact by placing the smaller

triangle inside the larger triangle.

40

40

70

70

70

70

The 40 degree angles are congruent.

40

70

70

70

70

The 70 degree angles are congruent.

40

40

70

70

70

70

70

The other 70 degree angles are congruent.

4

40

70

70

70

70

70

Find the length of the missing side.

50

?

30

6

40

8

This looks messy. Lets translate the two

triangles.

50

?

30

6

40

8

Now things are easier to see.

50

30

?

6

40

8

The common factor between these triangles is 5.

50

30

?

6

40

8

So the length of the missing side is?

Thats right! Its ten!

50

30

10

6

40

8

Similarity is used to answer real life questions.

- Suppose that you wanted to find the height of

this tree.

Unfortunately all that you have is a tape

measure, and you are too short to reach the top

of the tree.

You can measure the length of the trees shadow.

10 feet

Then, measure the length of your shadow.

10 feet

2 feet

If you know how tall you are, then you can

determine how tall the tree is.

6 ft

10 feet

2 feet

The tree must be 30 ft tall. Boy, thats a tall

tree!

6 ft

10 feet

2 feet

Similar figures work just like equivalent

fractions.

30

5

11

66

These numerators and denominators differ by a

factor of 6.

30

5

6

11

6

66

Two equivalent fractions are called a proportion.

30

5

11

66

Similar Figures

- So, similar figures are two figures that are the

same shape and whose sides are proportional.

Practice Time!

1) Determine the missing side of the triangle.

?

9

5

?

3

4

12

1) Determine the missing side of the triangle.

15

9

5

?

3

4

12

2) Determine the missing side of the triangle.

36

36

6

6

?

4

?

2) Determine the missing side of the triangle.

36

36

6

6

?

4

24

3) Determine the missing sides of the triangle.

39

?

?

33

?

8

24

3) Determine the missing sides of the triangle.

39

13

?

33

11

8

24

4) Determine the height of the lighthouse.

?

?

8

2.5

10

4) Determine the height of the lighthouse.

?

32

8

2.5

10

5) Determine the height of the car.

?

?

3

5

12

5) Determine the height of the car.

7.2

?

3

5

12