8th Grade - Dilation

Introduction

• To transform a figure (shape) means changing the size, location, and direction it faces.
• The figure before the transformation is called the pre-image. 
• The figure after the transformation is called the image. 
• If the pre-image is labeled as A, then the image would be labeled as A' (pronounced as A prime).
• There are four different types of transformation.

1. 1. Dilation
   2. Reflection
   3. Rotation
   4. Translation

Similarity

• The image after the dilation is similar to the pre-image.
• Two images are similar if:

• • They are of different sizes with proportional sides.
  • All angles are congruent.
  • They can be mapped on one another with a series of transformations.

• The symbol for similarity is ∼.

             

Dilation

• A dilation is a type of transformation that resizes (stretches or shrinks) the original figure.
• A dilation produces an image that is the same shape as the original but different in size.
• Dilation produces similar but not congruent images.
• It enlarges or reduces the size of the original figure (pre-image) with a scale factor. 
• A dilation needs a center point, also called as the center of dilation, and a scale factor.
• The center of dilation or the center point is a fixed point in the plane. 
• The scale factor is a ratio of the corresponding sides in a figure.                   

• A scale factor between 0 and 1 means the resulting image will be shrunk/reduced.
• A scale factor > 0 means the resulting image will be enlarged.
• A scale factor of 1 means the pre-image and image are congruent. 

 

Solved Examples

Question 1: If the scale factor of a dilation centered at the origin is 3, find the image of the point (2, 7).

Solution: If the scale factor is $k=3$, it follows the dilation rule $\left(x,y\right)\to \left(3x, 3y\right)$.

Hence, the dilated image of the point (2, 7) will be $\left(3\times 2, 3\times 7\right)\Rightarrow \left(6, 21\right)$.

Question 2: $△ABC$ is plotted on the coordinate grid. If the figure were dilated by a scale factor $\frac{5}{2}$ with the origin as the center of dilation, what are the coordinates of the vertices of $A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$.

Solution:

Pre-image	Image after dilation  $\left(scale factor=\frac{5}{2}\right)$
$A\left(-4, 8\right)$	$A^{\text{'}}\left(-10, 20\right)$
$B\left(8, 6\right)$	$B^{\text{'}}\left(20, 15\right)$
$C\left(4, -2\right)$	$C^{\text{'}}\left(10, -5\right)$

 

Question 3: $△A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ is the image of $△ABC$ under a dilation with a scale factor of 4 as shown in the figure below. Find the length of segment $\overline{A^{\text{'}}{C}^{\text{'}}}$.

Solution: $\text{scale factor}=\frac{\text{corresponding side length of image}}{\text{corresponding side length of pre-image}}$

$\text{scale factor}=\frac{A^{\text{'}}{C}^{\text{'}}}{AC}$

$4=\frac{A^{\text{'}}{C}^{\text{'}}}{4}$

$\Rightarrow A^{\text{'}}{C}^{\text{'}}=4\times 4=16 \text{units}$

Cheat Sheet

• Look for keywords: Stretch, Enlarge, Reduce, and Shrink.
• In dilation, the figure stretches or shrinks based on a scale factor.
• $\left(x,y\right) \to \left(kx,ky\right)$ where k is the scale factor.
• Dilated images are similar but not congruent.

Blunder Areas

• Differentiate between pre-image (A) and image (A').
• The scale factor in fractions does not always reduce the size of the figure.