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:

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:

^{6}⁄_{4} = ^{3}⁄_{2} because the cross products are equal; (6 x 2) = (3 x 4)

Example 2

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.

- Both figures are right, scalene triangles.
- The sides are proportional. The larger triangle’s sides are twice as large as the smaller triangle’s.
- 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 S_{k}, where k is a non-zero number that each coordinate is multiplied by. S_{3}means that the x- and y-values of each coordinate get multiplied by 3.

These figures are similar and have a size transformation of 3 (S_{3}).

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 Δ FDE are:F = (1,2) D = (-3,1) E = (-1,-1) Triangle DEF is shown in red. |

**More Dilation Examples**

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

**Try It!**

What are the coordinates of the second figure after a size transformation of 5 (S_{5})?

**Solution**

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

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

**Try It!**

Draw a similar a similar figure with a size transformation of 2 (S_{2})?

**Solution**

- Find the coordinates of figure JBXE
- Multiply these coordinates by 2
- 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.