IF you have an appropriate device, find at least one...

Warm-Up 5 minutes

IF you have an appropriate device, find at least

one example of where ratios and proportions are

used in the real world

Think about it… Before a building is built, an architect has to first

build a model of what they want the building to

look like. How do you think that their design is

transformed into the actual building?

Fashion? Theater? Music? Food?

Manufacturing? Architecture? Medicine?

homework

Chapter 6: Similarity 6.3: Using Similar Polygons

Objectives:

• If two figures are similar, how do you

find the length of a missing side?

Terms:

• Ratio:

• Scale Factor:

• Similar/Similarity:

y

xWritten as either or x : y

Ratio of the lengths for 2

corresponding sides.

Similar figures are figures that have the same

shape but not necessarily the same size.

To find the scale factor, what shape you are going TO over

where you are coming FROM: FROM

TO

Similar Figures

Similar Figures () mean “similar to”

Two figures are similar if two conditions are true:

1.

2. Corresponding SIDES are proportional

Corresponding ANGLES are congruent

After proving triangles are similar, we can prove that

their parts are congruent, too!

CASTC: Corresponding Angles of Similar Triangles are Congruent

CSSTP: Corresponding Sides of Similar Triangles are Proportional

Similarity

Example: Pentagon

PQRSTVWXYZ

List all the corresponding parts of the pentagon that are similar:

Angles: Sides:

*When you name similar figures, be sure to name

corresponding vertices IN THE SAME ORDER!!!

<P <V

<Q <W

<R <X

<S <Y

<T <Z

PQ VW

QR WX

RS XY

ST YZ

PT VZ

V

W

X Y

Z

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How to put vertices in order… You are given the following:

The sides are going to

match up so that the small

sides are proportional, the

middle sides are

proportional, and the big

sides are proportional.

1. Box the smallest number of each triangle with blue.

2. Circle the largest number of each triangle with green .

3. Triangle the number left over in red.

4. Now you can match up your vertices using color. Given CAT, you

know that since C is between blue and red, it will have to match up

with the vertex between blue and red on the other triangle, or G.

Scale Factor

If 2 polygons are similar, then the ratio of the lengths of two

corresponding sides is the scale factor of the similarity.

8

2

To find the scale factor of figure

#1 to figure #2:

#1 #2

4

1

8

2

1#

2#

So, the scale factor is 4

1

If the figure gets

smaller, then SF <1

If the figure gets larger,

then SF >1

If the figure stays the

same, then SF = 1

FROM

TO

Example Quad ABCD Quad EFGH. Find the

1)Scale Factor to ABCD and (2) values of x, y, and z.

Scale Factor:

Value of x, y, and z:

#1 #2

3

5

30

50:

2#

1#

x

50

3

5:

2#

1#

123

5:

2#

1# y

z

22

3

5:

2#

1#

x = 30 y = 20 z = 13.2

Always, Sometimes, Never…

For each statement, determine whether the statement

is always (AT), sometimes (ST), or never (NT) true:

a. Two rectangles are similar.

b. Two squares are similar.

c. A triangle is similar to a quadrilateral.

d. Two isosceles triangles are similar.

e. Two equilateral triangles are similar.

ST

AT

NT

ST

AT

Finding Sides of Similar

Figures Directions

1. Get a partner

2. Get out a sheet of paper between the two of

you. Fold the paper like a hot dog, and write

each person’s name at the top of a column.

3. Student B will copy the problem on his side of

the paper and work the problem while Student

A watches.

4. Repeat Step 2, reversing roles until all the

problems are finished.

5. Turn in, after checking answers . . .

Finding Sides of Similar

Figures Activity For the following problems, the figures are similar. Find the

value of each variable. The figures are not drawn to scale!

1. 2.

3. 4.

11.2

35/9

7.2 5.4

20 6

4.5

14 10.5

Finding Perimeters of

Similar Figures

• If two polygons

are similar, then

the ratio of their

perimeters is

equal to the

ratios of their

corresponding

side lengths.

Finding Perimeters of

Similar Figures- Example

Mario’s Pizzeria needs to rent a

bigger space. He currently has a

rectangular restaurant that is 60

feet long and 40 feet wide. He

would like to keep the same shape

of his restaurant but expand it so

that it will be 85 feet long. What is

the scale factor of the old restaurant

to the new one he wants?

What will be the perimeter of the new

restaurant?

12

17

60

85

FROM

TO

x = 283.333 ft

Chapter 6: Similarity 6.4 and 6.5: AA Similarity Postulate and

the SSS and SAS Similarity Theorems

Objectives:

• How do you prove triangles similar?

• 1. What appears to be true about the

triangles above?

• angles? _________ sides? _________

∆’s same size? ______ ∆’s same shape?

_____

• Similarity statement: _______~_________

• C_____

• T_____


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C

A

D T G

O

AA Similarity Postulate

If two angles of one triangle are _____________

to two angles of another triangle, then the two

triangles are ____________.

congruent

similar

AA Similarity Example 1. What do you know

about right angles?

2. You also know that

vertical angles are

congruent!

3. The triangles are

similar by AA

Similarity….BUT what is

the order of the

vertices?

ΔVWX ~ ΔZYX

• 2. What appears to be true about the

following triangles?

• angles? _________ sides? _________

∆’s same size? ______ ∆’s same shape?

_____


• A_____

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A

B

D C

F

E

6 16

9 24

AB

DE


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SAS Similarity Theorem

SAS: If two sides of one

triangle are proportional to two

sides of another triangle and

their included angles are

congruent, then the triangles

are similar.

• 3. What appears to be true about the triangles?

• angles? _________

• sides? __________

• ∆’s same size? ______

• ∆’s same shape? _____


• Compare smallest, largest . . .


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A

B

D 20

24

16

C F

E

9

7.5

6


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SSS Similarity Theorem

SSS: If three sides of one

triangle are proportional to

the three corresponding sides

of another triangle, then the

triangles are similar.

SSS Similarity Theorem

Example

DF

CB

DE

CA

EF

AB

22

4

EF

AB

23

6

DE

CA

23

6

DF

CB

Since the

sides all have

the same

scale factor,

they are

proportional.

SAS Similarity

Theorem Example

1. The only way these could be similar is by SAS

Similarity Theorem.

2. The small sides of each triangle have to be proportional,

and the big sides of each triangle have to be proportional

so that:

XZ

RT

XY

RS

3. Fill in the numerical values to make sure the sides are

proportional.

34.1

2.4

XY

RS3

9.1

7.5

XZ

RT

1. Draw a picture and label the

information.

2. Make sure the triangles are

similar – if you have parallel lines,

look for

____________________________. Alternate interior angles are congruent

3. Because you know the triangles are

similar by AA, you can set up your

proportions and solve for x.

DE

AE

DC

AB

)12(

)43(

8

4

x

x

)43(8)12(4 xx

x20

16

x5

4

4. Now that you know x, you can find

AE and DE:

AE =

DE =

5

264)

5

4(3

5

41212

5

4

Similar Figures – Proof

Example

Given: <B <C

Prove: ML

NM

MC

BM

Statements Reasons

1. <B <C 1. given

2. <1 <2 2. Vertical angles are congruent

3. Δ BNM ~ΔCLM 3. AA Similarity Postulate

4. Corresponding Sides of Similar

Triangles are Proportional

(CSSTP)

4.

ML

NM

MC

BM


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Class Examples

To estimate the height of a tree, a girl

scout sights the top of the tree in a mirror

that is 34.5 meters from the tree. The

mirror is on the ground and faces upward.

The scout is 0.75 meters from the mirror,

and distance from her eyes to the ground is

about 1.75 meters. How tall is the tree?

34.5 meters 0.75 meters

X meters

1.75/x = .75/34.5

60.375 = .75x

80.5 meters= x

1.75 meters

IF you have an appropriate device, find at least one...

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