Scale Factor & Dilations

Scale Factor

And

Dilations

What is a Dilation? What is Scale Factor? How do we graph an Image? How do we find the coordinates? How do we find Scale Factor?

​A dilation is a type of Transformation that changes the size of a figure without changing its shape.

Dilations can make a figure larger or smaller. For example, the first dilation below makes the triangle larger. The second dilation makes the triangle smaller.

​

In a dilation, the original figure is called the preimage. The dilated figure is called the image. Figures are dilated from a fixed point, called the center of dilation.

​When you dilate a figure, the scale factor determines how much smaller or larger the image will be.

• If the scale factor is between 0 and 1, the image is smaller than the preimage. Dilations with a scale factor between 0 and 1 are called reductions.

• If the scale factor is greater than 1, the image is larger than the preimage. Dilations with a scale factor greater than 1 are called enlargements.

When a figure is dilated, you can name each new point using the prime symbol, '. For example, the dilated image of point Q would be point Q', read "Q prime."

FUN FACT

​A preimage and its dilated image are similar, which means that corresponding angles have the same measure and corresponding sides are proportional. Because of this dilations can be classified as Similarity Transformations.

You can dilate figures on the coordinate plane. The origin is often used as the center of dilation. If you are dilating a figure centered at the origin, you can multiply the coordinates of the points in the preimage by the scale factor to determine the points in the image..

​

Dilations Centered at the Origin

Draw

Graph the image of

△EFG

after a dilation with a scale factor of

1/3

,

centered at the origin.

​Here is the correct answer.

​Let's see how we get to that answer.

​The scale factor of a dilation is the ratio of a length in the image to the corresponding length in the original figure.

This dilation is centered at the origin, so you can find the image by multiplying the x- and y-coordinates by the scale factor.

Multiply the coordinates of point E(–6,9) by 13. The image is E'(–2,3).

​Remember!

A dilation enlarges or reduces a figure.

​🌟

​

Dilations

• A dilation is a transformation in which a figure is enlarged or reduced with respect to a point called the center of dilation.

• With dilation, the preimage and image are similar.

• You can determine the scale factor by finding the ratio of the corresponding sides of the preimage and image.

​
Remember!

The rule for a dilation is
  (x,y)→(kx,ky)(x,y)→(kx,ky)  


k is your scale factor and we are multiplying. 

If |k| > 1, it is an ENLARGEMENT (stretch).
If 0<|k| < 1, it is a REDUCTION (shrink).

​🌟

Multiple Choice

Triangle ABC will be dilated by a scale factor of 1/2.  What will be the coordinates of C'?

• (-1, -2.5)

• (12, 6)

• (3, -1.5)

• (0, 1.5)

Multiple Choice

Dilate Point B by a scale factor of 1/2

• (1.5,-4)

• (-1.5,1)

• (-1,-1.5)

• (-2,-2)

Multiple Choice

Does the image show a Dilation? If yes, did the image enlarge, reduce or none of the above?

• Yes, enlarge

• Yes, reduce

• No

Multiple Choice

Did the image enlarge, reduce or none of the above?

• Enlarge

• Reduce

• None

Multiple Choice

If the triangle ABC is dilated using a scale factor of k=2, what will be the coordinates of A' after the transformation?(Click the picture to zoom in on it)

• A' (-6, 6)

• A'(1.5, -1.5)

Multiple Choice

Click to Enlarge Image and Answer the Question.

• A

• B

• C

• D

Multiple Choice

Click to Enlarge Image and Answer Below

• A

• B

• C

• D

​

Dilations create Similar Figures

• Corresponding angles are congruent

• Corresponding sides are proportional

• Since sides are proportional, we can write an equivalent proportion

​

What are corresponding parts?

• When part of an original figure matches up with part of a copy, we call them corresponding parts. 

• These could be points, segments, angles, or distances.

Multiple Choice

Find the scale factor

• Multiply by 1

• Multiply by 2

• Multiply by 3

• Multiply by 4

Multiple Choice

What is the scale factor from ΔDEF to ΔABC?

• 1/2

• 7/3

• 2/3

• 1/4

If an image gets smaller, the scale factor is LESS than 1.

​​REDUCTION

If an image gets bigger, the scale factor is GREATER than 1.

ELARGEMENT

Multiple Choice

Determine the scale factor for the similar figures from figure B to A.

• + 10

• x 3

• + 1/3

Multiple Choice

Determine the scale factor of the following similar figures from figure F to G.

• x 1/5 or x 0.2

• x 5

Multiple Choice

What is the scale factor from the small shape to the big shape?

• 2

• 1/2

Multiple Choice

Dilate Point B by a scale factor of 3:

• (12,0)

• (12,12)

• (0,12)

Poll

If your SF = 2.5, what type of dilation is it?

• Reduction

• Enlargement

Multiple Choice

If your SF = 0.8, what type of dilation is it?

• Reduction

• Enlargement

Multiple Choice

State the coordinate of the image of the given point G (-10,-6) under a dilation with center at the origin with the given scale factor of 1/2.

• G' (-20,-12)

• G' (-5,-3)

• G' (20,12)

Multiple Choice

What is the formula for scale factor?

• $$\frac{NEW}{OLD}$$

• $$\frac{OLD}{NEW}$$

Multiple Choice

Mr. Rivera takes a poster to STAPLES and tells them to use a scale factor 3.5 on the poster. What effect will a scale factor of 3.5 have on the original poster?

• Shrink

• Flip

• Turn

• Enlarge

Multiple Choice

A triangle has the following vertices: A(-1,1) B(6,-2) C(3,5). If the triangle undergoes a dilation with a scale factor of 4, what will be the coordinates of C'?

• A' (5,5) B'(23, -2) C'(15, 23)

• A' (-4,4) B'(24, -8) C'(12, 20)

Multiple Choice

Triangle PQR is shown. What are the coordinates of Q' when triangle PQR is dilated by a scale factor of 4 using the origin as center?

• Q' (16,20)

• Q' (20, 16)

• Q' (24, 9)

Multiple Choice

Triangle STU is a dilation of Triangle PQR. What is the scale factor in the dilation?

• 1/2

• 1.5

• 3/4

• 3

Multiple Choice

State the coordinate of the image of the given point B (4,9) under a dilation with center at the origin with the given scale factor of 2.

• B' (2 , 4.5)

• B' (8 ,18)

Multiple Choice

Which is a TRUE statement about dilations?

• A dilation will ALWAYS produce a similar figure.

• A dilation will ALWAYS enlarge a figure.

• A dilation will ALWAYS change the orientation of a figure.

Multiple Choice

The two rectangles are scale drawings. What is the measurement of x?

• 21

• 20

• 4