Characteristics of Dilation and Transformations

Key Concepts

• Understanding dilation
• Describing dilation
• Understanding similarity
• Describing similarity

Dilation and Transformations

Transformation, different transformation, rigid transformation. 

• Transformation is a change. 
• Translation is a transformation where the image or point slides across the plane. 
• Reflection is a transformation where the image flips across the line, and that line is called the line of reflection. 
• Rotation is a transformation that occurs around a fixed point, and that point is called the center of rotation. 
• Transformations in which there are no changes of size or shape after transformation are rigid transformations. 
• Translation, reflection, and rotation are examples of rigid transformation. 

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. 
• Center of dilation is the fixed point on the plane. 
• 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 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. 

• 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, 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. 

Key Concept Covered 

• Understand dilation 
• Describe dilation 
• Understand similarity 
• Describe similarity 

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.

What have we learned:

• What is a dilation?
• Dilation is a transformation that changes its size without changing the shape.
• What is a scale factor?
• The scale factor in dilation is the number multiplied by each side length of the original figure to get the length of the side of the image after transformation.
• What is similarity?
• Similarity of two figures can be defined as the figures that undergo changes in size without changing the shape of the figure.
• What is the criteria for identifying two similar triangles?
• In similar triangles, there is a proportionality between the corresponding sides and corresponding angles are congruent.

Concept Map