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Mastering Dilations and Similarity

Smart Worksheet · Grade 8 · English

Mastering Dilations and Similarity

Activity 1: Real-World Similarity

Observe the image of the Russian Nesting Dolls below. These dolls are designed so that each smaller doll is a scaled version of the larger one.
A set of Russian Nesting Dolls showing proportional scaling.
A set of Russian Nesting Dolls showing proportional scaling.

Task: In geometry, we call this relationship similarity. If the largest doll is 12 inches tall and the next doll is 9 inches tall, what is the scale factor (

) of the reduction? Show your work below.


Activity 2: Labeling a Dilation

A dilation involves several key components. Study the diagram below and match the correct term to each numbered part.
Dilations Labeling Diagram
Dilations Labeling Diagram

Word Bank:
  • Scale Factor Projection Line
  • Pre-image (Original Figure)
  • Center of Dilation
  • Image (Dilated Figure)
Labels:
  1. 
  1. 
  1. 
  1. 


Activity 3: Calculating Missing Dimensions

Triangle

has been dilated to create Triangle

. The two triangles are similar (

).
Two similar triangles with labeled sides
Two similar triangles with labeled sides

Task:
  1. Determine the scale factor
    
    using the corresponding sides
    
    and
    
    .
  1. Use the scale factor to calculate the missing length
    
    (side
    
    ).
Your Calculation:


Activity 4: Navigating Similarity

Use the flowchart below to determine if two geometric figures can be classified as similar.
Similarity Process Flowchart
Similarity Process Flowchart

Task: You are given two rectangles. Rectangle A has sides of 4 and 8. Rectangle B has sides of 6 and 12. All angles in both rectangles are 90°. Following the flowchart, explain step-by-step why these rectangles are similar.


Activity 5: Coordinate Plane Dilation

On the coordinate grid below, Triangle A is the pre-image.
Triangle A on a coordinate grid
Triangle A on a coordinate grid

Task:
  1. Write the coordinates for the vertices of Triangle A: (____, ), (, ), (, ____).
  1. Dilate Triangle A using a scale factor of
    
    with the origin (0,0) as the center.
  1. Plot the new image (Triangle A') on the grid and list the new coordinates: (____, ), (, ), (, ____).


Activity 6: Scale Factor Challenge

A square with a side length of 5 cm is dilated. The new square has a perimeter of 40 cm. What is the scale factor (

) of the dilation?
A)

B)

C)

D)



Activity 7: Defining the Center

In your own words, describe what happens to the Center of Dilation during a transformation. Does its position change? Why or why not?


Activity 8: Sequences of Transformations

Triangle

is at

,

, and

. It undergoes a sequence of transformations:
  1. A dilation with
    
    centered at the origin.
  1. A translation 5 units down.
Task: What are the final coordinates of vertex

?


Activity 9: Similarity Statements

If

, complete the following corresponding parts:
  1. 
    ______
  1. 
    ______
  1. 


Activity 10: Comparison Organizer

Complete the T-Chart below to summarize the differences between a Dilation Enlargement and a Dilation Reduction.
Name:
Date:
Enlargement (k > 1)
Reduction (0 < k < 1)
Effect on size:
Effect on size:
Scale Factor Example:
Scale Factor Example: