NACS g.srt.0SimilarityRightTrianglesOV 1 of 2 3/4/2014

Unit Overview Similarity, Right Triangles & Trigonometry Unit G.SRT

This unit has four distinct stages to it and they each build on one another.

FOUNDATION – DILATIONS

While this level of the house has only one objective it is VERY IMPORTANT one. Dilation leads use directly into

similarity. This is where the basic properties of dilation are taught and the concepts of how angles and parallel

lines connect us to similarity. This is also where the essential skills of setting up ratios and solving for

unknown sides are developed. While there is only one objective this objective should take over a week to

properly develop.

1ST STORY – SIMILARITY

Similarity is the BIG CONCEPT of this unit. These objectives help to develop new ideas in a triangle about

proportions and ratios. Students learn to prove similarity and they learn and use proportional parts when

working with parallel lines and the connection of all of this to the geometric mean and special right triangles.

NACS g.srt.0SimilarityRightTrianglesOV 2 of 2 3/4/2014

2ND STORY – TRIGONOMETRY

Trigonometry at its core is similarity. With it, understanding of the fixed ratios for all similar triangles is gained

and it leads to the development of a conceptual understanding for all right triangles. The ratios of Sine, Cosine

and Tangent are specifically named, discussed and used. If similarity is built correctly, then trigonometry

happens quite naturally.

THE ATTIC – LAW OF SINES AND COSINES (HONORS)

While this level is an honors only level, it too is a natural progression. After working with only right triangles

and their trigonometric relationships we want to explore oblique triangles. These laws help to expand the

ability to solve for sides and angles into any triangle.

NACS g.srt.1PropertiesChararctericsDilationOV 1 of 1 3/4/2014

Overview Properties and Characteristics of Dilation Unit G.SRT.1

G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of a line segment is longer or shorter depending on the ratio given by the scale factor.

Similarity is directly connected to the dilation transformation and so it is essential to build a strong understanding of the properties and characteristics of dilation. Students need to look at the effects of dilation on lines and point as well as different scale factors and centers.

Dilations are the foundation of the entire unit…. Take the time to really develop the properties.

(1) The student will determine the properties of dilation.

(2) The student will be able to dilate when the center of dilation is in, on or out of the shape.

(3) The student will be able dilate when given a center of dilation and a scale factor.

(4) The student will be able to determine the center of dilation and the scale factor from a diagram.

(5) The student will be able to use the dilation coordinate rules for dilations using any center of dilation.

The purpose of learning dilations is to prepare the students for similarity. Students need to feel comfortable with dilations that create enlargements and reductions. Dilations centered at a vertex create the classic overlapping triangle diagram that shows up regularly in similarity. The definition of dilation also helps establish relationships about angles and parallel lines later. DON’T SHORT CUT THIS AREA. Spend time developing a strong basis in dilation skill and understanding.

1 – Focus on the definition of dilation. It will help establish relationships in similarity. 2 – Enlargements & Reductions. Students struggle with getting ratios correct and this focus will help students learn the proper way to write a scale factor. 3 – Work Backwards. To test understanding start with a dilation and have students determine its scale factor and center. If students can work backwards, then they understand the process.

NACS g.srt.1PropertiesCharacteristicsDilationsNotes 1 of 4 3/4/2014

Notes Properties and Characteristics of Dilation Unit G.SRT.1

CONCEPT 1 – Dilation Properties So what actually happens when a shape is dilated? The length of each side of the image is equal to the length of the corresponding side of the pre-image multiplied by the scale factor, k;

A’B’ = k AB, B’C’ = k BC and A’C’ = k AC.

This dilation has a scale factor of 1:3.

AB: A’B’ 4: 12 1:3

BC: B’C’ 5.5: 16.5 1:3

AC: A’C’ 5:15 1:3

DEFINITION

A dilation with center O and a scale factor of k is a transformation that maps every point P in the plane to point P’ so that the following properties are true.

1. If P is NOT the center O, then the P’ lies on OP .

The scale factor k is a positive number such that k = 'OP

OP

and k 1.

2. If P is the center point O, then P = P’. The center of dilation is the only point in the plane that does not move.

'OP

OP> 1

0 < 'OP

OP < 1

PROPERTIES

DILATION PROPERTIES - A dilation is NOT an isometric transformation so its properties differ from the ones we

saw with reflection, rotation and translation. The following properties are preserved between the pre-image and its

image when dilating:

Angle measure (angles stay the same)

Parallelism (things that were parallel are still parallel)

Collinearity (points on a line, remain on the line)

Distance IS NOT preserved!!!

After a dilation, the pre-image and image have the same shape but not the same size.

16.5 cm

15 cm

5.5 cm

5 cm

12 cm

4 cm

C'

B'

A'

A

B

C

O = P = P'

Enlargement P'

O

P

Reduction

P' P

O

NACS g.srt.1PropertiesCharacteristicsDilationsNotes 2 of 4 3/4/2014

CONCEPT 2 - Verify experimentally the properties of dilations given by a center and a scale

factor: a dilation takes a line passing through the center of dilation unchanged.

Dilation of AB when the center of dilation is point O on the line with scale factor of k, k > 1.

Dilation of AB when the center of dilation is point O on the line with scale factor of k, 0 < k < 1.

PRE-IMAGE ( AB ) IMAGE ( AB = ' 'A B )

, ( ) ( )O kD AB AB

The result of dilating AB by a scale factor of k > 1

with the center of dilation that is on the AB is simply

AB . The dilation maps all points either on to OA or

OB which are the opposite rays that form AB when O is between A and B.

PRE-IMAGE ( AB ) IMAGE ( AB = ' 'A B )

, ( ) ( )O kD AB AB

The result of dilating AB by a scale factor of 0 < k < 1 with the center of dilation that is on the

AB is simply AB . The dilation maps all points

either on to OA or OB which are the opposite

rays that form AB when O is between A and B.

B

A

O

B'

A'

B

A

O

B

A

O

B'

A'

B

A

O

original original

Longer

dilation

Shorter

dilation

NACS g.srt.1PropertiesCharacteristicsDilationsNotes 3 of 4 3/4/2014

CONCEPT 3 - Verify experimentally the properties of dilations given by a center and a scale

factor: a dilation takes a line not passing through the center of the dilation to a parallel line.

PRE-IMAGE ( AB ) IMAGE ( ' 'A B )

The center of dilation is point O (not on AB ) and the scale factor is k > 1.

Angles are preserved in dilation,

thus OBA OB’A’ and because these corresponding angles are congruent,

|| ' 'AB A B .

It is critical to understand that dilations create parallel lines between

ALL pre-image and image corresponding segments and lines.

O

B

A

B'

A'

O

B

A

NACS g.srt.1PropertiesCharacteristicsDilationsNotes 4 of 4 3/4/2014

CONCEPT 4 – The coordinate rule for dilation.

There is not much to discover here because the coordinate rule is provided in the definition,

, ( , ) ( , )O kD x y kx ky

When O is the origin

This was already discovered intuitively in G.CO.2 when investigating isometric and non-isometric transformations. It was

discovered that if one or two variables was multiplied by a value other than 1 or -1 there was no longer an isometry.

The difference between the dilation and the stretch was also established. Dilations must have the same value multiplied

to both variables, whereas the stretch has different values.

DILATION STRETCH

,5( , ) (5 ,5 )OD x y x y

1,3

1 1( , ) ( , )

3 3OD x y x y

, 3( , ) ( 3 , 3 )OD x y x y

Using S as the notation for stretch:

( , ) (2 ,3 )S x y x y

Indicates a two-way stretch. Double horizontal and triple vertical.

2( , ) ( ,5 )

3S x y x y

( , ) (8 , )S x y x y

Indicates a one-way horizontal stretch.

It is obvious that for a dilation to maintain its proportionality of sides,

the two variables must be multiplied by a constant value, k, known as the scale factor.

NACS g.srt.1PropertiesCharacteristicsDilationAssess 1 of 3 3/4/2014

Assessment Properties and Characteristics of Dilation Unit G.SRT.1

1. Which of the following is a dilation?

A) ( , ) ( 4, 3)T x y x y B) ( , ) ( , )T x y y x

C) ( , ) (2 ,2 )T x y x y D) ( , ) (5 ,3 )T x y x y

2. Which of the following is a non-isometric transformation?

A) Rotation B) Dilation C) Reflection D) Translation

3. Which is not an example of dilation?

A) The pupil of your eye reducing in size due to the bright sun.

B) Using a graphics program to alter a pictures size from 8 by 10 to 4 by 5.

C) Standing in front of a circus fun house mirror that makes you very tall.

D) Using a shrink gun to make you one-fourth the size that you are now.

4. If we ,2OD then the correct diagram would be:

Original A) B) C) D)

5. Determine the scale factor that best suits the provided diagram (O is the center of

dilation).

A) 2 B) 1

2 C)

1

3 D) - 1

6. Determine the scale factor of the given dilation from point O?

A) 1 : 2 B) 2 : 1

C) 2 : 5 D) 5 : 2

m

O

m'm

O

m'm

O

m'm

O

B'

A'

A

B

O

2.5 cm

5 cmB'

C'O

B

C

m’

O

NACS g.srt.1PropertiesCharacteristicsDilationAssess 2 of 3 3/4/2014

7. Given ,2 ( , ) '( 4,10)OD P x y P then P(x, y) is

A) P(-8, 20) B) P(-2 ,5) C) P(-4, 12) D) P(0, 20)

8. Determine whether the dilation is an enlargement or a reduction and then determine the scale factor.

Place the scale factor in the most reduced form (no decimals).

a) Enlarge or Reduce

_____ : _____

b) Enlarge or Reduce

_____ : _____

c) Enlarge or Reduce

_____ : _____

d) Enlarge or Reduce

_____ : _____

e) Enlarge or Reduce

_____ : _____

f) Enlarge or Reduce

_____ : _____

g) Enlarge or Reduce

_____ : _____

h) Enlarge or Reduce

_____ : _____

9. Dilate the following. (O is the origin).

a) ,2(2, 1) (____ , ____)OD

b) , 3( 2,4) (____ , ____)OD

c) 1,3

(3,12) (____ , ____)O

D

d) ,4

5 12( , ) (____ , ____)4 8

OD e) 3,

4

(8, 4) (____ , ____)O

D

f) ,2.5(10,4) (____ , ____)OD

I'

I

6 cm

2 cm

B'C'

O

B

C 6 cm

4 cm

4 cm

7 cm

C'B'

D'

D

BC

1 cm

2 cm

A'

O

A

13.5 cm4.5 cm

C'

B' B

C

1 cm

7 cm

6 cm

3 cm

B'

P'

C'

O

C

P

B

I' I1 cm

2 cm

A

O

A'

NACS g.srt.1PropertiesCharacteristicsDilationAssess 3 of 3 3/4/2014

Answers:

1) C

2) B

3) C

4) D

5) B

6) B

7) B

8) a) Reduce, 3 : 1

b) Reduce, 4 : 1

c) Reduce, 3 : 2

d) Enlarge, 2 : 3

e) Reduce, 3 : 1

f) Enlarge, 1 : 3

g) Reduce, 3 : 2

h) Reduce, 3 : 2

9) a) (4, -2)

b) (6, -12)

c) (1, 4)

d) (5, 6)

e) (-6, 3)

f) (25, 10)