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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