DAY 71 – VERIFYING DILATIONS

INTRODUCTION

We have already encountered dilations in our previous lessons on transformations. A dilation produces an image that is the same shape as the pre-image but same or different size, therefore, it is worth noting that dilation is not a rigid transformation in general.

In this lesson, we will base our discussion on the effects of dilations on line segments as part of verifying dilations. We will discover what happens to a line segment when dilation is performed with the center of the dilation on the segment and with the center at another point, not on the segment. We will also relate the dilation of a line segment to the ratio given by the scale factor.

VOCABULARY

1. Dilation

A transformation that enlarges or reduces the pre-

image while preserving shape by moving all points

along a ray passing through a fixed point called the

center of dilation.

2. Center of dilation

A fixed point in the plane where all the points on

the pre-image are moved along a ray passing

through this point.

3. Scale factor of the dilation

The ratio of the length of corresponding sides on

the image and the pre-image

4. Collinear points

Points that lie on the same straight line.

Before we verify the effects of dilations on line

segments, it is important to recall some key

properties of dilations:

1. The pre-image and the image have the same

shape but different sizes, unless the scale factor is

equal to 1

2. Angles are mapped to congruent angles

3. Parallel lines are mapped to parallel lines

4. The ratios of corresponding sides on the pre-

image and the image are equal.

5. The ratios of corresponding line segments are

equal to the scale factor

6. Line segments are mapped to line segments

with a given scale factor

We can now discuss the following concepts about

dilations and lines and line segments:

1. A dilation leaves a line passing through the

center of dilation unchanged, that is, the

image lies on the same line.

If an entire line is dilated with the center of

dilation on the line, the line remains unchanged.

After a dilation, image points and corresponding

pre-image points all lie on the straight line to the

center of dilation. This concept applies to line

segments on plane figures too.

Consider line AB (AB) shown below. Note that the

center of dilation, O is on AB. If a dilation with scale

factor 2 is performed on points A and B, they will be

mapped to points A′ and B′ respectively.

Note that the points A,B, A′ and B′ are collinear.

If we dilate the entire AB about point O, by using

point A as the pre-image point then point A′ should lie

on the line through point O and point A which still

lies on AB. This means that since point O is located on

AB then the images of all points on AB will lie on AB.

OA B B′A′

We can therefore conclude that, if a line is dilated

in such a way that the center of dilation lies on the

line, the dilation does not change the line, that is,

we get the same line.

Note that in case of line segments, the pre-image

and the image both line on the same line.

In the figure above, AB and its image A′B′ both lie

on the same line, AB after a dilation with scale

factor 2 about the point O.

OA B B′A′

2. A dilation takes a line not passing through

the center of the dilation to a parallel line

We can also dilate a line when the center of

dilation does not lie on the line.

Consider the diagram below which shows a

dilation of PQ with point O as the center of dilation

and a scale factor of 2. Note that the point O is not

on PQ.

The points O, P and P′ are collinear.

Similarly, the points O, Q and Q′ are collinear.

Dilations preserve angle measures, therefore,

∠OPQ ≅ ∠OP′Q′.

OP

Q

P′

Q′

If a pair of lines is intersected by a transversal

and the resulting corresponding angles are

congruent, then the two lines are parallel, thus

PQ ∥ P′Q′.

Based on the illustration above, we can conclude

that a dilation takes a line not passing through

the center of the dilation to a parallel line.

In case of line segments on plane figures, the

line segments are mapped to parallel line

segments. In the figure above, PQ and its image

P′Q′ are parallel.

3. The dilation of a line segment is longer or

shorter in the ratio given by the scale factor.

If the scale factor, 𝑘 is greater than 1:

(a) the resulting image is larger than the pre-

image.

(b) the line segments, which are the sides of the

image will be longer than the corresponding sides

of the pre-image.

(c) it is referred to as an enlargement.

(d) the pre-image lies between the center of dilation

and the image.

This illustrated below using ΔXYZ and its image

ΔX′Y′Z′ after a dilation about the point O, scale

factor, 𝑘 > 1.

X

Y Z

X′

Y′ Z′

O

According to the figure above;

X′Y′ > XY, Y′𝑍′ > YZ and X′Z′ > XZ

If we use a scale factor greater than 1, say, 3, we

shall have:

X′Y′ ≅ 3XY, Y′𝑍′ ≅ 3YZ and X′Z′ ≅ 3XZ

This shows that the lengths of the line segments

on the image will be three times longer than those

of the corresponding segments on the pre-image.

The scale factor is taken as the ratio of the length

of any side on the image to the length of the

corresponding side on the object, that is X′Y′

XY=Y′Z′

YZ=X′Z′

XZ

If the scale factor lies between 0 and 1:

(a) the resulting image is smaller than the pre-

image.

(b) the line segments, which are the sides of the

image will be shorter than the corresponding sides

of the pre-image.

(c) it is referred to as an reduction.

(d) The image lies between the center of dilation

and the pre-image.

If the scale factor is equal to 1, the pre-image

and the image are congruent

This illustrated below using ΔXYZ and its image

ΔX′Y′Z′ after a dilation about the point O, scale

factor 𝑘 such that 0 < 𝑘 < 1.

X

Y Z

X′

Y′ Z′

O

According to the figure above;

X′Y′ < XY, Y′𝑍′ < YZ and X′Z′ < XZ

If we use a scale factor 𝑘 such that, 0 < 𝑘 < 1, say, 1

4, we shall have:

X′Y′ ≅1

4XY, Y′𝑍′ ≅

1

4YZ and X′Z′ ≅

1

4XZ

Example

If the length of one side on a triangle is 4.4 inches,

what would be the length of the corresponding side

after the triangle is dilated by a scale factor of 3

2

about a point O located away from the triangle?

State whether the side and its corresponding side

on the image will be parallel or not.

Solution

Scale factor =Length of side on the image

Length of corresponding side on the object

Length of side on the image = Scale

factor × Length of corresponding

side on the object

=3

2× 4.4 = 6.6 𝑖𝑛𝑐ℎ𝑒𝑠

The side and its corresponding side on the image will

be parallel.

HOMEWORK

Use a ruler to dilate AB about the center C with a

scale factor of 2. Estimate the locations of points A

and B on the line. State whether A′B′ lies on AB or

not.

CA B

ANSWERS TO HOMEWORK

A′B′ lies on AB.

CA B B′A′

THE END