similarity, right triangles, and trigonometry

Clusters:Clusters:1. Understand similarity in terms of 1. Understand similarity in terms of

similarity transformationssimilarity transformations2. Prove theorems involving 2. Prove theorems involving

similarity.similarity.3. Define trigonometric ratios and 3. Define trigonometric ratios and

solve problems involving right triangles.solve problems involving right triangles.4. Apply trigonometry to general 4. Apply trigonometry to general

triangles.triangles.

Similarity, Right Triangles, Similarity, Right Triangles, and Trigonometryand Trigonometry

Learning TargetLearning Target

1. I can define dilation.

2. I can perform a dilation with a given center and scale factor on a figure in the coordinate plane.

Connection to previews Connection to previews lesson…lesson…

• Previously, we studied rigid transformations, in which the image and preimage of a figure are congruent. In this lesson, you will study a type of nonrigid transformation called a dilation, in which the image and preimage of a figure are similar.

DilationsDilations

A dilationdilation is a type of transformation that enlarges or reduces a figure but the shape stays the same.

The dilation is described by a scale factorscale factor and a center of center of dilationdilation.

DilationsDilations

Side Original

Side Newk

The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage. It describes how much the figure is enlarged or reduced.

The dilation is a reduction if k < 1 and it is an enlargement if k > 1.

• •C C

P

Q

R

P

Q

R

P´

Q´

R´

P´

Q´

R´

3

6

2

5

Reduction: k = = =36

12

CPCP

Enlargement: k = =52

CPCP

PQR ~ P´Q´R´, is equal to the scale factor of the dilation.

P´Q´PQ

Constructing a DilationConstructing a Dilation

Examples of constructed a dilation of a triangle.

Steps in constructing a Steps in constructing a dilationdilationStep 1: Construct ABC on a coordinate

plane with A(3, 6), B(7, 6), and C(7, 3).18

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5 10 15 20 25 30 35O

A B

C

Steps in constructing a Steps in constructing a dilationdilationStep 2: Draw rays from the origin O through A, B, and C. O is the center of dilation.

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Steps in constructing a Steps in constructing a dilationdilationStep 3: With your compass, measure the

distance OA. In other words, put the point of the compass on O and your pencil on A. Transfer this distance twice along OA so that you find point A’ such that OA’ = 3(OA). That is, put your point on A and make a mark on OA. Finally, put your point on the new mark and make one last mark on OA. This is A’.

Steps in constructing a Steps in constructing a dilationdilationStep 3:

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A'

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A B

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Steps in constructing a Steps in constructing a dilationdilationStep 4: Repeat Step 3 with points B and C.

That is, use your compass to find points B’ and C’ such that OB’ = 3(OB) and OC’ = 3(OC).

Steps in constructing a Steps in constructing a dilationdilationStep 4:

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A'

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A B

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B'A'

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C'

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Steps in constructing a Steps in constructing a dilationdilation You have now located three points, A’, B’,

and C’, that are each 3 times as far from point O as the original three points of the triangle.

Step 5: Draw triangle A’B’C’.

A’B’C’ is the image of ABC under a dilation with center O and a scale factor of 3. Are these images similar?

Steps in constructing a Steps in constructing a dilationdilationStep 5:

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Questions/ Observations:Questions/ Observations:

Step 6: What are the lengths of AB and A’B’? BC and B’C’? What is the scale factor?

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•AB = 4•A’B’= 12•BC = 3•B’C’= 9

Questions/ Observations:Questions/ Observations:

Step 7: Measure the coordinates of A’, B’, and C’.

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B'A'

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ImageA´(9, 18)B´(21, 18)C´(21, 9)

Questions/ Observations:Questions/ Observations:

Step 8: How do they compare to the original coordinates?

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B'A'

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Pre-image A(3, 6) B(7, 6) C(7, 3)

ImageA´(9, 18)B´(21, 18)C´(21, 9)

P(x, y) P´(kx, ky)

In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).

Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).

Use the origin as the center and use a scale factor of . How does the perimeter of

the preimage compare to the perimeter of the image?

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A

A´1

1O

B´

C´D´ B

CD

x

y

•

SOLUTIONBecause the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor.

A(2, 2) A´(1, 1)B(6, 2) B´(3, 1)C(6, 4) C ´(3, 2)D(2, 4) D´(1, 2)

From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6.

In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).

Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of . How does the perimeter of the preimage compare to the perimeter of the image?

12

SOLUTION

A

A´1

1O

B´

C´D´ B

CD

x

y

•A preimage and its image after a dilation are similar figures. Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

Example 3Example 3

Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation.

Is this a reduction or an enlargement?

Assignment/HomeworkAssignment/Homework

Work with a partner in the classwork on “Constructing Dilation”

Homework:

Answer Guided Practice page 510 #12 to 15.