NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
Lesson 1
M2
Date__________________
Lesson 1: Introductions to Dilations
GEOMETRY
Learning Target
 I can create scale drawings of polygonal figures
 I can write scale factor as a ratio of two sides and determine its numerical value
A dilation is a transformation whose preimage and image are similar. Thus, a dilation is a similarity
transformation. It is not, in general, a rigid motion.
Every dilation has a center and a scale factor n, n > 0. The scale factor describes the size change from the
original figure to the image.
A dilation with center R and scale factor n, n > 0, is a transformation with the following properties:
2 Types of Dilations
A dilation that creates a larger image is called an
A dilation that creates a smaller image is called a
enlargement. When the image is an enlargement the
reduction. • When the image is reduced that the scale
scale factor is greater than 1. The image expands.
factor is between 0 and 1. The image contracts.
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
Lesson 1
M2
Date__________________
GEOMETRY
The Notation for Dilation is: 𝐷𝑂,𝑟 where O: ________ of ______________ and r: _________ _________
Definition: A dilation is a rule (a function) that moves points in the plane a specific distance along the ray that
originates from a center 𝑂.
What determines the distance a given point moves?
1. If r > 1 the dilation will push the point _______________ from the center.
2. If r = 1 the dilation will keep the point ________________ from the center of dilation
3. If 0 < r < 1 the dilation will pull the point _________________ the center.
Finding the Scale Factor
Example 1. Find the scale factor
The ratio of corresponding sides
is the scale factor (n) of the dilation.
Example 2. The dashed line figure is a dilation image of the solid-line figure. D is the center of dilation. Tell
whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.
Example 3. The image of ∆𝐸𝐷𝐹 after a dilation of scale factor
k centered at point D is ∆𝐺𝐷𝐻 , as shown in the diagram
below. What is the scale factor? ( give as a ratio of two sides)
Quick Write Is a dilation a rigid motion? Explain
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
M2
Date__________________
Drawing Dilation Images
GEOMETRY
Example 3. Draw the dilation image 𝛥𝐵′𝐶′𝐷′ = 𝐷(2,𝑋) (𝛥𝐵𝐶𝐷)
https://www.youtube.com/watch?v=wbDktALdi_s&feature=youtu.be
1
Example 4. Given center 𝑂 and triangle 𝐴𝐵𝐶, dilate the figure from center 𝑂 by a scale factor of 𝑟 = 4. Label
the dilated triangle 𝐴′𝐵′𝐶′
Connect the center of dilation O to all vertices
Measure the length of segments
OA =_____, OB=_____, OC=_______
Apply the dilation rule for each length and calculate
OA’ =_____, OB’=_____, OC’=_______
Remember your dilated points A’,B’,C’ will be on the
same rays that connect the center of dilation to each
vertices
Measure each length and connect the new points,
label the image A’B’C’
Find the following ratios
𝑂𝐶′
𝑂𝐶
= _____
Write the scale factor as a ratio of sides
𝑂𝐵′
𝑂𝐵
= ____
𝑂𝐴′
𝑂𝐴
= ___
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
M2
Date__________________
Example 4 (continued)
GEOMETRY
1
A line segment 𝐴𝐵 undergoes a dilation of 𝑟 = 4. What will the length of the image segment A’B’?
Angle ∠𝐶𝐵𝐴 measures 78°. After a dilation, what will the measure of ∠𝐶′𝐵′𝐴′ be? How do you know?
Example 5 Given center 𝑂 and triangle 𝐴𝐵𝐶, dilate the triangle from center 𝑂 with a scale factor 𝑟 = 3.
Measure the length of segments
OA =_____, OB=_____, OC=_______
Apply the dilation rule for each length and
calculate
OA’ =_____
, OB’=_____ , OC’=_______
Find the following ratios
𝑂𝐶′
𝑂𝐶
= ____
𝑂𝐵′
𝑂𝐵
= ____
𝑂𝐴′
𝑂𝐴
Set up an extended proportion of the
corresponding lengths
=
Using a ruler measure AB= ________ BC = _________ and AC = __________
Using a ruler measure A’B’ = ________ B’C’ = _________ A’C’ = ___________
Set up an extended proportion of the corresponding side-lengths
=
=
=
Lesson Summary
There are two properties of a scale drawing of a figure:
1. Corresponding angles are equal _________ in measurement
2. Corresponding lengths, sides are proportional __________ in measurement.
=
= ____
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
M2
Date__________________
Lesson 1: Introductions to Dilations
GEOMETRY
Classwork
Exercise 1. The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center
of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of
the dilation
a.
b.
Exercise 2: Find the scale factor
As a ratio of sides
Numerical Value
Exercise 3. The image of
after a dilation of scale factor k centered at
point A is
, as shown in the diagram below. What is the scale factor?
Exercise 4. In the diagram below,
factor k with center E.
is the image of
Which ratio is equal to the scale factor k of the dilation?
after a dilation of scale
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
M2
Date__________________
GEOMETRY
Exercise 5. A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true?
1)
The area of the image is nine times the· area of the original triangle.
2)
The perimeter of the image is nine times the perimeter of the original triangle.
3)
The slope of any side of the image is three times the slope of the corresponding side of the original triangle.
4)
The measure of each angle in the image is three times the measure of the corresponding angle of the original
triangle.
Exercise 6. In the diagram below,
factor?
As a ratio of sides
is the image of
after a dilation centered at the origin. What is the scale
Numerical Value
1
Exercise 7) Create a scale drawing/ dilation of the figure below about center 𝑂 with scale factor 𝑟 = 2.
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
M2
Date__________________
Lesson 1: Introductions to Dilations
GEOMETRY
Homework
1. Create a scale drawing of the figure below about center 𝑂 and scale factor 𝑟 = 3.
Measure the length of OA : _________
Measure the length of OA' : _________
What is the ratio of OA to OA’?__________________
Write the scale factor as a ratio of two sides
𝒎∠𝑨 = 𝟏𝟕°, 𝒎∠𝑩 = 𝟏𝟑𝟒° , 𝒎∠𝑪 = 𝟐𝟐°, 𝒎∠𝑫 = 𝟐𝟑°
What will be the measures of 𝒎∠𝑨′ = _________ 𝒎∠𝑩′ = __________, 𝒎∠𝑪′ = ________, 𝒎∠𝑫′ = _____
NYS COMMON CORE MATHEMATICS CURRICULUM
Name__________________
2. Dilate circle 𝐴, from center 𝑂 at the origin by scale factor 𝑟 = 3.
M2
Lesson 1
Date__________________
GEOMETRY
1
4 . Use the ratio method to create a scale drawing about center 𝑂 with a scale factor of 𝑟 = 4.
Give the proper notation of the Dilation: _______________