5-7 similar figures and proportions course 2 warm up warm up problem of the day problem of the day...

5-7 Similar Figures and Proportions

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Presentation – Six LessonsPresentation – Six Lessons

Warm UpFind the cross products, then tell whether the ratios are equal.

Course 2


1. 166

, 4015

2. 38

, 1846

3. 89

, 2427

4. 2812

, 4218

240 = 240; equal

216 = 216; equal

504 = 504; equal

138 = 144; not equal

Problem of the DayEvery 8th telephone pole along a road has a red band painted on it. Every 14th pole has an emergency call phone on it. What is the number of the first pole with both a red band and a call phone?

56

Course 2


Lesson 1EQ: How can I determine if two figures are similar?

Course 2


Vocabulary Words

similar

corresponding sides

corresponding angles

Insert Lesson Title Here

Course 2


Course 2


Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similar figures have the same shape but not necessarily the same size.

Course 2


SIMILAR FIGURESTwo figures are similar if• The measures of their corresponding angles are

equal. • The ratios of the lengths of the corresponding

sides are proportional.

Course 2


Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles.

82◦

Corresponding angles

D

E

F

Corresponding sides

A

B

C

82◦

55◦43◦ 55◦43◦

Course 2

5-7 Math Dictionary: SYMBOLS

When naming similar figures, list the letters of the corresponding vertices in the same order. In the previous table ∆ABC ~ ∆DEF.

Math Tip:

∆ABC

AB

║ and

~

Course 2


A side of a figure can be named by its endpoints, with a bar above.

AB

Without the bar, the letters indicate the length of the side.

Reading Math

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

Example 1: Determining Whether Two Triangles Are Similar

Course 2


A C

B

10 in

4 in7 in

D

E

F

16 in 28 in

40 in

AB corresponds to DE.

BC corresponds to EF.

ABDE

=? BC

EF=? AC

DF4

167

281040

14

14

14

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Step 1: Write ratios using the corresponding sides.

Step 2: Substitute the length of the sides.

Step 3: Simplify each ratio.

=? =?

AC corresponds to DF.

=? =?

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

Check It Out: Example 2

Course 2


A C

B

9 in

3 in 7 in D

E

F

9 in 21 in

27 in

AB corresponds to DE.

BC corresponds to EF.

ABDE

=? BC

EF=? AC

DF39

721

927

13

13

13

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Write ratios using the corresponding sides.

Substitute the length of the sides.

Simplify each ratio.

=? =?

AC corresponds to DF.

=? =?

Lesson 2

EQ: How can I determine if figures are similar based on their angle measure?

Course 2


Tell whether the figures are similar.

How can I determine if these shapes are similar?

Course 2


Yes….The corresponding angles of the figures have equal measure.

60°

60°

F

A

E

D

BC

*remember – the sum of the interior angles of a triangle = 180°

Try One:


Course 2


Tell whether the figures are similar. (Notice the shapes are turned)

1. similar59°

59°

35°35°

86°86°

Try another:


Course 2


Tell whether the figures are similar.

2.

not similar

119°

55°

107°

79°

107°

80°

135°

38°

Lesson 3

EQ: How can I determine the scale factor of similar figures?

Course 2


Scale Factor:

The ratio of the lengths of corresponding sides in similar figures

Course 2


EXAMPLE 1

The figures below are similar

How can I determine the scale factor of similar figures?

Course 2


9

12

3

4

EXAMPLE 2

A ~ B

How can I determine the scale factor of similar figures?

Course 2


5

2.5

2

1

A

B

Lesson 4

EQ: What is the relationship between the scale factor, side lengths, perimeter, and area?

Course 2


The relationship between scale factor, side lengths, perimeter, and area...

10 ft

3 ft6 ft

11 ft5.5 ft

5 ft

Figure A BRatio/Scale Factor

Corresponding Sides

Side Lengths

(feet)

Perimeter

Area

*The scale factor tells you the ratio of corresponding side lengths and the ratio

of the perimeters

*The scale factor SQUARED tells you the

ratio of the areas

Lesson 5

EQ: How can I determine missing side lengths of similar figures?

Course 2


Find the unknown length in similar figures.

Example 1: Missing Side Lengths

ACQS

= ABQR Step 1: Write a proportion using corresponding sides.

1248

= 14w

Step 2: Substitute lengths of the sides.

12 · w = 48 · 14 Step 3: Cross multiply and divide.12w = 672

12w12

= 67212

w = 56

QR is 56 centimeters.

Divide each side by 12 to isolate the variable.

Course 2

5-8 Using Similar Figures



Course 2


A B

C D

10 cm

12 cm

Q R

S T

24 cm

ACQS

= ABQR

Write a proportion using corresponding sides.

1224

= 10x

Substitute lengths of the sides.

12 · x = 24 · 10 Find the cross product.

12x = 240 Multiply.

12x12

= 24012

x = 20

QR is 20 centimeters.


Find the unknown length in similar figures.x

The inside triangle is similar in shape to the outside triangle. Find the length of the base of the inside triangle.


Course 2


Let x = the base of the inside triangle.

82

=12x

8 · x = 2 · 128x = 24

8x8

= 248

x = 3The base of the inside triangle is 3 inches.

Write a proportion using corresponding sidelengths.

Find the cross products.Multiply.


Example 3: Measurement Application

Example 4

The rectangle on the left is similar in shape to the rectangle on the right. Find the width of the right rectangle.


Course 2


3 cm

6 cm12 cm

Let w = the width of the right rectangle.

612

= 3w

6 ·w = 12 · 3

6w = 36

6w6

= 366

w = 6

The right rectangle is 6 cm wide.

Write a proportion using correspondingside lengths.

Find the cross products.Multiply.


?

Ticket-out-the-door

Find the unknown length in each pair of similar figures.


Course 2


1.

2.

Ticket-out-the-door

Find the unknown length in each pair of similar figures.


Course 2


3. The width of the smaller rectangular cake is 5.75 in. The width of a larger rectangular cake is 9.25 in. Estimate the length of the larger rectangular cake.

Lesson 6

EQ: How can I use shadow math to find missing side lengths?

Course 2


Course 2


Step 1: Label Corresponding Parts.

Step 2: Write a Proportion.

Step 3: Cross multiply anddivide.x

5m1m

1.5m

Example 1: Missing Side Lengths

Additional Example 2: Estimating with Indirect Measurement

Course 2


City officials want to know the height of a traffic light. Estimate the height of the traffic light.

27.2515

= 48.75h

Step 1: Label Corresponding Parts.

95

≈ 49h

Step 2: Write a Proportion

9h ≈ 245

The traffic light is about 30 feet tall.

27.25 ft

48.75 ft

h ft 2715

≈ 49h

Step 3: Cross multiply.

h ≈ 27 Multiply each side by 9 to isolate the variable.


Course 2


The inside triangle is similar in shape to the outside triangle. These are called NESTED triangles. Find the height of the outside triangle.

514.75

= h30.25

Write a proportion.

Use compatible numbers to estimate.

13

≈ h30

Simplify.

1 • 30 ≈ 3 • h

The outside triangle is about 10 feet tall.

14.75 ft

30.25 ft

h ft

515

≈ h30

30 ≈ 3h Multiply each side by 5 to isolate the variable.

5 ft

Cross multiply.

10 ≈ h

Classwork

• Problem 5.1(pg.78-79)

• Problem 5.2 (pg. 80-81)

• Problem 5.3 (pg. 82-83)

5-7 similar figures and proportions course 2 warm up warm up problem of the day problem of the day...

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lengths of corresponding

similar figurestwo figures

ratio of corresponding

similar figuresexample

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similar ifthe measures

pair of triangles

matching angles