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Clusters 1. Understand similarity in terms of

similarity transformations 2. Prove theorems

involving similarity. 3. Define trigonometric

ratios and solve problems involving right

triangles. 4. Apply trigonometry to general

triangles.

Similarity, Right Triangles, and Trigonometry

Learning Target

- I can define dilation.
- I can perform a dilation with a given center and

scale factor on a figure in the coordinate plane.

Connection to previews 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.

Dilations

- A dilation is a type of transformation that

enlarges or reduces a figure but the shape stays

the same. - The dilation is described by a scale factor and a

center of dilation.

Dilations

- 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 lt 1 and it is an

enlargement if k gt 1.

P

P

6

5

P

P

2

3

Q

Q

R

C

C

R

Q

R

Q

R

Constructing a Dilation

- Examples of constructed a dilation of a triangle.

Steps in constructing a dilation

- Step 1 Construct ?ABC on a coordinate plane

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

Steps in constructing a dilation

- Step 2 Draw rays from the origin O through A,

B, and C. O is the center of dilation.

Steps in constructing a dilation

- Step 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 dilation

- Step 3

Steps in constructing a dilation

- Step 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 dilation

- Step 4

Steps in constructing a dilation

- 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 ABC.
- ?ABC 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 dilation

- Step 5

Questions/ Observations

- Step 6 What are the lengths of AB and AB? BC

and BC? What is the scale factor?

- AB 4
- AB 12
- BC 3
- BC 9

Questions/ Observations

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

Image A(9, 18) B(21, 18) C(21, 9)

Questions/ Observations

- Step 8 How do they compare to the original

coordinates?

P(x, y) ? P(kx, ky)

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

?

Image A(9, 18) B(21, 18) C(21, 9)

In a coordinate plane, dilations whose centers

are the origin have the property that the image

of P(x, y) is P(kx, ky).

SOLUTION

Because the center of the dilation is the origin,

you can find the image of each vertex by

multiplying its coordinates by the scale factor.

C

D

A

D

B

A(2, 2) ? A(1, 1)

B(6, 2) ? B(3, 1)

B

C(6, 4) ? C (3, 2)

D(2, 4) ? D(1, 2)

In a coordinate plane, dilations whose centers

are the origin have the property that the image

of P(x, y) is P(kx, ky).

SOLUTION

From the graph, you can see that the preimage has

a perimeter of 12 and the image has a perimeter

of 6.

C

D

A

D

B

A preimage and its image after a dilation are

similar figures.

B

Therefore, the ratio of the perimeters of a

preimage and its image is equal to the scale

factor of the dilation.

Example 3

- Determine if ABCD and ABCD are similar

figures. If so, identify the scale factor of the

dilation that maps ABCD onto ABCD as well as

the center of dilation.

Is this a reduction or an enlargement?

Assignment/Homework

- Work with a partner in the classwork on

Constructing Dilation - Homework
- Answer Guided Practice page 510 12 to 15.