Dilations

Here you will learn about dilations, including how to enlarge a 2D shape by a scale factor and how to describe dilation geometry in detail.

Students will first learn about dilations as part of geometry in 8 th grade.

What are dilations?

Dilations are a type of transformation that change the size of a figure without changing its shape. This transformation involves resizing a figure by a specific scale factor relative to a fixed point, often called the center of dilation.

Dilations have real life functions, such as changing the size of photographic prints or pictures in documents.

• Enlarging a shape by a scale factor greater than \bf{1} will make the shape bigger.

For example,

Shape A has been enlarged by scale factor 2 to make shape B.

The corresponding angles are identical, but the length of each side in shape B is double the size of the original shape.

You need to multiply the original lengths by the scale factor to find the lengths of the enlarged shape.

• Reducing a shape by a scale factor between \bf{0} and \bf{1} will make the shape smaller. This is often written as a fraction and is known as a fractional scale factor.

For example,

Shape A has been reduced by scale factor \cfrac{1}{2} to make shape B.

The corresponding angles are identical, but the length of each side in shape B is half the size of the original shape.

The object is the name of the original shape.

The image (or dilation image) is the name of the shape after it has been translated.

When an object is enlarged or reduced, the object and the image are similar shapes.

Step-by-step guide: Scale factor

Center of dilation

The center of dilation places the enlargement or reduction in a specific place.

To use a center of dilation, you need to draw straight lines from the center of dilation through the vertices of the original shape. These are called ray lines.

For example, here triangle ABC has been enlarged by scale factor 2 about a center of dilation point O. The new triangle is labelled A’B’C’.

The lengths of the sides of the new triangle are double the lengths of the sides of the original triangle.

The pairs of corresponding sides are parallel lines.

The angles in the two shapes are the same.

The triangles are similar triangles.

Step-by-step guide: Center of dilation

What are dilations?

Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8 – Geometry (8.G.A.3)
  Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

• High School – Geometry – Similarity, Right Triangles, & Trigonometry (HS.G.SRT.A.1a)
  A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

• High School – Geometry – Similarity, Right Triangles, & Trigonometry (HS.G.SRT.A.1b)
  The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

How to solve problems involving dilations

In order to solve problems involving dilations:

1. Understand the problem.
2. Apply the dilation transformation or solve for the desired information.

Dilations examples

Example 1: use a scale factor to enlarge a shape

Enlarge the shaded shape by scale factor 2.

1. Understand the problem.

The problem is asking you to draw a second shape as an enlargement of the original shape given using a scale factor of 2.

2Apply the dilation transformation or solve for the desired information.

You can apply the dilation transformation here by multiplying the original lengths by the scale factor to find the lengths of the enlarged shape. It is easier to start with horizontal or vertical lines.

Example 2: calculating a scale factor

Shape A has been enlarged to make shape B. Calculate the scale factor.

Example 3: with a center of dilation on a grid

Enlarge the shaded shape with scale factor 3 about the center of dilation.

Apply the dilation transformation or solve for the desired information.

Use the ray lines to help you enlarge the shape.

Example 4: with a center of dilation on a coordinate plane

Enlarge the shaded shape by scale factor 2 about the point (1,2) .

Apply the dilation transformation or solve for the desired information.

First, you’ll need to plot the center of dilation on the coordinate grid. Then draw ray lines from the center of dilation through the vertices of the original shape. Use the ray lines to help you enlarge the shape.

Example 5: find a center of dilation

Shape A has been enlarged to make shape B. Find the center of dilation.

Apply the dilation transformation or solve for the desired information.

Find pairs of corresponding vertices and draw ray lines going through the points.

Then, plot the center of dilation accordingly.

Example 6: negative scale factor (ADVANCED)

Enlarge the shaded shape with scale factor – 2 about the center of dilation.

Apply the dilation transformation or solve for the desired information.

Draw ray lines from the center of dilation through the vertices of the original shape.

Extend the ray lines backwards through the center of dilation, as this is where the new points will go.

Teaching tips for dilations

• Start with basic polygons like triangles and rectangles. This makes it easier for students to see the effects of dilation before moving on to more complex figures.

• Use interactive tools or video lessons to visually demonstrate how changing the scale factor and center of dilation affects the figure.

• Educators can pose questions like, “What happens if the center of dilation is at a vertex?” or “How does the figure move if the center is outside?”

• After completing problems, ask students to explain the dilation changes. This reinforces their understanding and ability to articulate the concept.

Easy mistakes to make

• Inaccurate drawings
  Use a sharp pencil and make use of the grid lines to help you to be accurate.

• Using diagonal lines
  Diagonal lines can be tricky to enlarge, so it is best to use horizontal and vertical lines.

• Misunderstanding the origin
  The origin of a coordinate grid has the coordinates (0,0), where the x -value and y -value are both zero. It is commonly denoted as O. It is used often as the center of dilation.

• Describing \bf{2} or more transformations
  If you are asked to give a single transformation – make sure it is a single transformation, not 2 or more. Also, make sure that you state the type of transformation and give full details.
  
  For example,
  For enlargements state scale factor and the coordinates of the center of dilation.

Related transformations lessons

• Transformations
• Rotations
• Translation math
• Reflection in math

The next lessons are

• Geometric proofs
• Area
• 3D shapes

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