Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation

A dilation is a transformation that produces an image that is of the same shape and different sizes.

Dilation that creates a larger image is called enlargement.

• Dilation that creates a smaller image is called reduction.
• The center of dilation is the fixed point on the plane.
• The scale factor is the ratio between the length of the original image to the transformed image.
• Dilation is not a rigid transformation as it preserves only the shape.
• Zooming can be given as an example of dilation.

Describing Dilation

Dilation of Scale Factor 2

The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ (6, 6),
C’ (8, 2).

D: (X,Y)                  (2x , 2y)              A(1,2), B(3,3), C(4,1)

Characteristics of Dilation

Each angle of the figure and its image remains the same.

• The Midpoint of the sides of the figure remains the same as the midpoint of the dilated shape.
• Parallel and perpendicular lines in the figure remain the same as the parallel and perpendicular lines of the dilated image.
• The image remains the same.
• If the scale factor is greater than 1, the image stretches.
• If the scale factor is between 0 and 1, the image shrinks.
• If the scale factor is 1, then the original image and the dilated image are congruent.

Similarity

Two figures are said to be similar if their corresponding angles are congruent, and the ratio of the length of the corresponding sides is proportional.

The ratio of the perimeters is the same as the scale factor of similar triangles.

The scale factor for similar figures is a: b and the ratio of their areas is the scale factor squared: 
a² : b².

The following figures are similar because the ratio of the length of corresponding sides is proportional, and the scale factor is 2.

In similar triangles, corresponding angles are congruent.

All corresponding angles are equal

Check whether the following figures are similar; if so, describe the similarity.

AB = 72, BC = 48, AC = 84

HG = 12, GF = 8, HF = 14

AB/HG = 72/12 = 6                               

BC/GF = 48/8 = 6                

AC /HF = 84/14 = 6

The corresponding sides of the figure are proportional, so the triangles are similar.

Exercise:

1. State whether a dilation with the given scale factor is an enlargement or a reduction.

a.  Scale factor = 2,              b.  Scale factor =1/8,            c.  Scale factor = 5/4

2. Graph the image of rectangle KLMN after dilation with a scale factor of 4, centered at the origin.

3. Draw a dilation of scale factor 3.

4. Find the vertices of the dilated image and describe it as enlargement or reduction.

D : (x, y) (2x, 2y)

5. Scale factor of A to B is 1:9. Find the missing perimeter.

6. The scale factor of two regular octagons is 4:1. Find the ratio of their perimeters and the ratio of their areas.

7. Find the missing length. The triangles in each pair are similar.

8. Solve for x. The triangles in each pair are similar.

Concept Map: