Dilations

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Working through the lesson below with your child identify the characteristics of similar figures and create dilations of a given figure on a Cartesian plane.

Learning Takeaways: After this lesson, students will be able to:

  • Identify the characteristics of similar figures
  • Create a dilation of a given figure on a Cartesian plane

Make sure your child is familiar with the vocabulary below:

  • Similar Figures: Figures that are the exact same shape but different sizes.  The sides of each figure are proportional, the angle measures are equal.
  • Proportional: Two values that increase or decrease in size relative to each other

Similar Figures

This section will help your child to identify the characteristics of similar figures.

Similar Figures

  • Are the same shape
  • Have proportional sides
  • Have the same angle measures

Look at the examples below.

Example 1:

two similar rectangles

These rectangles are similar. They are the same shape. The sides are proportional. The angles all measure 90°.

Determining Proportional Sides

The lengths of the sides are proportional because lengths of each side of figure B are half of the corresponding side in figure A.

You can also create a proportion:

64 = 32 because the cross products are equal; (6 x 2) = (3 x 4)

Example 2

two similar triangles

These triangles are scalene, which means each angle and side is a different measure.  The angles of these similar triangles are 90°, 22.62°, and 67.68°.

These triangles are similar.

  1. Both figures are right, scalene triangles.
  2. The sides are proportional. The larger triangle's sides are twice as large as the smaller triangle's.
  3. The angles are equal.

Creating a Dilation

This section will help your child to create a dilation of a given figure on a Cartesian plane.

Make sure your child is familiar with the vocabulary below:

  • Dilation: A dilation is also called a size transformation. It is written as Sk, where k is a non-zero number that each coordinate is multiplied by. S3 means that the x- and y-values of each coordinate get multiplied by 3.
two similar polygons

These figures are similar and have a size transformation of 3 (S3).

 

The coordinates of ΔABC are:
A = (2,4)  B = (-6,2)  C = (-2,-2)
 
Triangle DEF has a size transformation of 0.5 (S
0.5).  That means you multiply the coordinates of ΔABC by 0.5.

The coordinates for ΔFDE are:
F = (1,2)  D = (-3,1)  E = (-1,-1)

Triangle DEF is shown in red.

dilation on cartesian plane

More Dilation Examples

Work through the two questions below with your child. The answers are shown but to try find them without looking!

square on cartesian plane

Try It!
What are the coordinates of the second figure after a size transformation of 5 (S
5)?

Solution

  1. Find the coordinates of the figure ABCD shown on the Cartesian Plane.
  2. The coordinates are: A= (-4,3), B = (4,3) , C = (-4,-4), D = (4,-4)
  3. Multiply both numbers in the coordinate pair by 5

The coordinates of the similar figure are: (-20 , 15); (20 , 15); (-20, -20); (20 , -20)

 

polygon on cartesian plane

Try It!
Draw a similar a similar figure with a size transformation of 2 (S
2)?

Solutionpreloaded image

  1. Find the coordinates of figure JBXE
  2. Multiply these coordinates by 2
  3. Redraw the figure using the new coordinates

The coordinates of the similar figure are: (2 , 6); (6 , -4); (-8, 0); (-6 , 0)

Click here to see the similar figure on the Cartesian plane (or mouse over the image)

Similarity Worksheets

Click the links below and get your child to try the similarity worksheets that will allow practice with questions based on what is shown above.