Dilations and similar figures

Dilations and similar figures

by Coach Miller

​Get out paper & pencil for notes & space to work out problems.

​Lesson Information

​standards: MGSE9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.

a. The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.

MGSE9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

​Learning targets: use the definition of dilation or the dilation coordinate rule to perform dilations on the coordinate plane.

• evaluate shapes on the coordinate plane to determine the scale factor and center of a dilation that has occurred.

• check the lengths of various sides to determine if they are proportional.

Essential Questions

​Describe what happens when you perform a dilation. 

What data was used to evaluate the scale factor of the dilation?

Multiple Choice

Two objects that are the same shape but not the same size are _______.

• Congruent
• Vertical
• Similar
• Complementary 

Multiple Choice

All angles in similar figures are congruent.  

• True
• False

​solving for a missing side length in a similar figure

​Find corresponding side lengths

​set up a proportion to solve

​

​4.5 = x

​7.5 5

Multiple Choice

Solve for X.

• 2 ft
• 3 ft
• 4 ft
• 6 ft

Multiple Choice

The pair of figures is similar. Find the missing side.

• x = 20
• x = 12
• x = 5
• x = 4

Multiple Choice

The triangles are similar, solve for the question mark.

• 8
• 12.5
• 18
• 24

Multiple Choice

The __________ is the ratio of a length of the new figure to the corresponding length on the original figure.

• reduction
• enlargement
• scale factor
• center of dilation

Multiple Choice

A dilation produces a similar figure

• True

• False

Multiple Choice

When the scale factor of a dilation is less than one, the dilation is a reduction (gets smaller).

• True

• False

Multiple Choice

An image is dilated at a scale factor of 1.5. This means the dilation is an enlargement and the dilated image will be bigger.

• True

• False

Multiple Choice

If your SF = 0.8, what type of dilation is it?

• Reduction

• Enlargement

Multiple Choice

Under a dilation, which scale factor will result in a congruent image?

• -2
• 0
• 1
• ¹/₂

Multiple Choice

What is the formula for scale factor?

• $$\frac{new}{old}$$

• $$\frac{old}{new}$$

Multiple Choice

What is the scale factor from the smaller quadrilateral to the larger?

• scale factor = 4
• scale factor = 10
• scale factor = 3
• scale factor = 2

Multiple Choice

What is the scale factor? from large shape to small shape

• 1

• 2

• 1/2

• -2

Multiple Choice

What is the scale factor from ΔABC to ΔDEF?

• 1/2

• 7/3

• 2

• 10/3

Multiple Choice

What is the scale factor of the dilation? 

• 1/3

• 2/3

• 3/2

• 3/1

What is the scale factor (keep in mind the total distance of the original figure)? ​

​

​

What is YT?

​

​

What is YY'?

​

​

​An example worth writing down

Multiple Choice

What is the scale factor of the dilation of triangle DEF? 

• 3/10

• 3/7

• 7/3

• 7

Multiple Choice

if RD = 15, what is RD'?

• 3 units

• 5 units

• 6 units

• 9 units

Multiple Choice

What is x, the distance between points S' and S? First find the whole length of QS'. Then, subtract out SQ

• x = 2.25 units

• x = 2.5 units

• x = 4.25 units

• x = 4.5 units

Multiple Select

Trapezoid ABCD was dilated to create trapezoid A'B'C'D'. Which statements are true about the trapezoids? Select three options

• The  length of AD is 8 units

• The length of A'D' is 4 units

• The image is larger than the pre-image

• Sides CD and C'D' both have the same slope, 2

• The scale factor is 1/2

​Determining coordinates based off scale-factor

​1st ensure that the center of dilation is the origin

2nd identify coordinates

3rd multiply each part of the coordinate by the scale factor 

​

​M (-4,6) --> (-4*1.5, 6*1.5) = M'(-6, 9)

​N (2, 6) --> (2*1.5, 6*1.5) = N'(3,9)

​If they ask you to identify the original coordinates, go backwards by dividing by the scale factor

Multiple Choice

How do you write the algebraic notation when the scale factor is 3?

• (x, y) goes to (1/3x, 1/3y)
• (x, y) goes to (-3x, -3y)
• (x, y) goes to (x + 3, y + 3)
• (x, y) goes to (3x, 3y)

Multiple Choice

Which algebraic expression could be the rule for a reduction?

• (x, y)→(2x, 2y)
• (x, y)→(5/2 x, 5/2 y)
• (x, y) →(0.5x, 0.5y)
• (x, y)→(5x, 5y)

Multiple Choice

Dilate Point B by a scale factor of 3:

• (12,0)
• (12,12)
• (4,3)
• (0,12)

Multiple Choice

The rule for the dilation of quadrilateral QRST to the image Q'R'S'T' is  D_O,0.5(x, y) → (0.5x, 0.5y).

What are the coordinates of Q', if Q(0, 2)?

• (0,1)

• (1,0)

• (0,4)

• (4,0)

Multiple Choice

State the coordinate of the image of the given point B (-10,-6) under a dilation with center at the origin with the given scale factor k = 1/2.

• (5,3)
• (20,12)
• (-20,-12)
• (-5,-3)

Multiple Choice

State the coordinate of the image of the given point B (4,9) under a dilation with center at the origin with the given scale factor k = 2.

• (2,4.5)
• (-8,-18)
• (8,18)
• (9,4)

Multiple Choice

Given △ABC, use a dilation with the center at the origin to make a similar triangle with side lengths three times as large. What are the coordinates of C' of the image? 

• (–6, 3)

• (-3,-6)

• (3,-6)

• (6,-3)

Multiple Choice

Triangle XYZ was dilated using the rule D_O,0.25 (x, y) → (0.25x, 0.25y).

The image is shown in the diagram. If the coordinates of Z' are (-2,-1) What are the coordinates of Z the pre-image?

• (-8,-4)

• (-8,-2)

• (-2,-1)

• (-0.25, -0.25))

Multiple Choice

Quadrilateral QRST is dilated and translated to form similar figure Q'R'S'T'. What is the scale factor for the dilation?

• 1

• 1/3

• (5,0)

• (-1,2)

Multiple Choice

Which graph has figures that can undergo a similarity transformation to justify that they are similar?

Multiple Choice

Triangle ABC was dilated and translated to form similar triangle A'B'C'.

What is the scale factor of the dilation? 

• 1/5

• 2/5

• 5/2

• 5/1

Multiple Choice

Which composition of similarity transformations would map polygon ABDC to polygon A'B'D'C'?

• a dilation with a scale factor less than 1 and then a reflection

• a dilation with a scale factor greater than 1 and then a rotation

• a dilation with a scale factor greater than 1 and then a translation

• a dilation with a scale factor less than 1 and then a translation

Multiple Choice

Which composition of similarity transformations maps polygon ABCD to polygon A'B'C'D'? 

• a dilation with a scale factor less than 1 and then a reflection

• a dilation with a scale factor less than 1 and then a translation

• a dilation with a scale factor greater than 1 and then a reflection

• a dilation with a scale factor greater than 1 and then a translation

Multiple Choice

What is the scale factor used to dilate the small square to the larger square given their areas?

Hint: first find each side length

• 4
• 16
• 3
• 12

Multiple Choice

What is the scale factor used to dilate the small square to the larger square given there perimeters? Hint: first find each side length

• 3
• 4
• 12
• 1/3