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

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.

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 Δ The coordinates for Δ Triangle DEF is shown in red. |

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)

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.