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

 Dilation
 Reflection
 Rotation
 Translation
Similarity
 The image after the dilation is similar to the preimage.
 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 (preimage) 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 preimage 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: $\u25b3ABC$ 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:
Preimage  Image after dilation $\left(scalefactor=\frac{5}{2}\right)$ 
$A\left(4,8\right)$  ${A}^{\text{'}}\left(10,20\right)$ 
$B\left(8,6\right)$  ${B}^{\text{'}}\left(20,15\right)$ 
$C\left(4,2\right)$  ${C}^{\text{'}}\left(10,5\right)$ 
Question 3: $\u25b3{A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ is the image of $\u25b3ABC$ 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: $\text{scalefactor}=\frac{\text{correspondingsidelengthofimage}}{\text{correspondingsidelengthofpreimage}}$
$\text{scalefactor}=\frac{{A}^{\text{'}}{C}^{\text{'}}}{AC}$
$4=\frac{{A}^{\text{'}}{C}^{\text{'}}}{4}$
$\Rightarrow {A}^{\text{'}}{C}^{\text{'}}=4\times 4=16\text{units}$
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 preimage (A) and image (A').
 The scale factor in fractions does not always reduce the size of the figure.
 Fiona Wong
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