Geometry notes: Dilations
1
We should start by revising scale factors and proportional relationships;
PROPORTION -- A proportion is when two ratios are equal to each other. A proportion allows us to correctly
adjust values in the same relationship. For example, if you are trying to mix water and juice powder in a 2:3
ratio but you want 10 cups of water used because you have lots of friends coming over for a swimming party.
You would need to set up a proportion as shown below.
2 10

3
x
2:3 to 10:x where x would be the number of cups of juice powder
that you need to put into the mix. To solve a problem like this we
cross multiply and solve.
2x = 30
x = 15
Cross multiplying creates 2x = 3(10) which is an easy equation to
solve. The value of 15 tell us that to keep the same ratio of 2:3 you
need to use 10:15.
The process of cross multiplication is used with all proportions. It is just a shortcut for a number of other
operations.
In the above example we demonstrate how to determine if two ratios are equivalent. If two ratios are
equivalent or have the same proportion they balance each other out when we cross multiply them out.
Equality of Cross Products For any numbers a and c and any nonzero numbers b and d,
a c
 if and only if ad = bc.
b d
Let’s look at four examples below.
1. Is the ratio 3:4 the same as 7:9?
2. Is the ratio 1.5:2 the same as 13.5:18?
NO, these ratios are not proportional!!
YES, these ratios are proportional!!
3. What value for x would create the same
proportion?
4. What value for x would create the same proportion?
Geometry notes: Dilations
2
5. Solve each proportion using cross products.
6
3
6
x


x x6
5 15
21  x 2

x
1
6(x – 6) =
3x
5x = 6(15)
x = 18
21 - x = 2x
21 = 3x
6x – 36 =
3x
7=x
3x = 36
x =12
We can use proportions and ratios to solve lots of different problems. Here are a few examples.
6. The ratio of boys to girls in the class is 4:3. If there are 20 boys in the class, how many girls are there?
boys 4 20
 
girls 3
g
4g = 3(20)
g = 15 There are 15 girls in the class.
7. The ratio of boys to girls in the class is 1:3. If there are 28 students in the class, how many girls are there?
Method #1
Method #2
girls 3
g
 
total 4 28
1x + 3x = 28
4g = 3(28)
4x = 28
g = 21 There are 21 girls in the class.
x=7
So there are 3(7) = 21 girls in the class.
8. A triangle has angles in a ratio of 1:2:3, what is the size of each angle?
1x + 2x + 3x = 180
6x = 180
x = 30
Angles are 30:60:90
9. What are the possible names for the quadrilateral with sides 3:1:3:1?
This tells us that opposite sides are equal and that all four sides are not equal…. Thus this quadrilateral
could be a parallelogram or a rectangle.
10. A recipe instructs the cook to use 4 cups of water for each 3 cups of powder. If you used 10 cups of
water, how much powder should be added?
water
4 10
 
powder 3
p
4p = 3(10)
p = 7.5
Add 7.5 cups of powder.
Geometry notes: Dilations
3
11. The soccer club has a 3:2 of juniors to freshman. If the club has 45 students attend that are freshman
and juniors, how many juniors are there?
juniors 3
j
 
total
5 45
5j = 3(45)
j = 27 There are 27 juniors in the club.
12. The sides of a triangle are in the ratio of 3:8:7. If the perimeter of the triangle is 63 cm, how long is the
shortest side?
3x + 8x + 7x = 63
18x = 63
x = 3.5
The smallest angle is 3(3.5) = 10.5 cm long.
Do
1. Solve each proportion using cross products.
a)
b)
c)
20  x 6

x
4
1
6

x x  15
3
x

5 15
d)
4
x2

12 2 x  13
x = _______
x = _______
x = _______
x = _______
e)
f)
g)
h)
x  1 x 1

6
x
x = _______
3
9

4 x7
x = _______
x 9 2

x
5
x = _______
x
4

5 16
x = _______
Geometry notes: Dilations
4
2. Solve the following problems. (Show work)
a) The ratio of seniors to juniors in the Chess Club is
2:3. If there are 24 juniors, how many seniors are in
the club?
b) A 15 foot building casts a 9 foot shadow. How tall
is the building that casts a 30 ft shadow at the same
time
c) A picture is 3 in. wide by 5 in. high was enlarged so
that the width was 15 inches. How high is the
enlarged picture
d) Cameron has been eating 2 dollar menu burgers
every week (7 days). At that rate, how many
hamburgers will he in 4 weeks?
e) A triangle’s three angles are in the ratio of 5:7:8.
What is the measure of the smallest angle?
f) A 6 foot high school boy casts a shadow of 24
inches. At the same time of day a girl at the
elementary school park casts a shadow of 14 inches.
How tall is she (in feet)?
Geometry notes: Dilations
5
3. Solve the following problems. (Show work)
a) The ratio of two supplementary angles is 4:5. Find
the measures of each angle.
b) The ratio of two complementary angles is 2:3. Find
the measures of each angle.
c) A 3 foot stick is broken into two pieces. The ratio
of the two pieces is 5:7. How big are the two pieces?
d) Is the largest angle acute, right or obtuse in a
triangle that has angles measures in ratio, 2:3:4?
e) Points A, B, C, and D are placed in alphabetical
order on a line so that AB = 2BC = CD. What is the
ratio BD : AD?
f) Points A, B, and C are placed in alphabetical order
on a line so that 3AB : AC. What is the ratio of AB :
BC?
g) Two numbers are in ratio 7 : 3. The sum of the two
numbers is 36. What is the largest number?
h) Three numbers are in the ratio of 2 : 5 : 3. If the
largest number is 65. What is the smallest number?
Geometry notes: Dilations
6
Activity
Ratios, Proportions and Currency.
One of the most applicable places where we find the use of ratios and proportions in the ‘Real World’ is in the
exchanging of currency. If you have ever travelled to a foreign country you have obviously experienced
currency exchanges. Currency has different values and $100 USD (United States Dollars) is not worth 100
Mexican Pecos or 100 Euros – currencies exchange at a given ratio which fluctuates daily. In the table below
each currency is being compared to 1 USD. The chart below shows some countries and how they compare
with our currency. The below table is from Sept. 27 th, 2013.
A stronger currency to USD has a lower value than 1.
A weaker currency to the USD has a higher value than 1.
A equal currency to the USD has an equal value of 1.
DZD
AOA
ARS
AMD
AUD
BZD
BWP
BRL
GBP
BGN
MMK
KHR
CAD
CLP
CNY
COP
CRC
HRK
CUP
CZK
DKK
DOP
ANG
EGP
AED
ETB
EUR
XAU
GTQ
HTG
HUF
Algerian Dinar
Angolan Kwanza
Argentine Peso
Armenian Dram
Australian Dollar
Belizean Dollar
Botswana Pula
Brazilian Real
British Pound
Bulgarian Lev
Burmese Kyat
Cambodian Riel
Canadian Dollar
Chilean Peso
Chinese Yuan Renminbi
Colombian Peso
Costa Rican Colon
Croatian Kuna
Cuban Peso
Czech Koruna
Danish Krone
Dominican Peso
Dutch Guilder
Egyptian Pound
Emirati Dirham
Ethiopian Birr
Euro
Gold Ounce
Guatemalan Quetzal
Haitian Gourde
Hungarian Forint
81.31999969
97.61935675
5.699999809
405.6186293
1.073306858
1.959800005
8.591065039
2.25165
0.619640113
1.444485
969
4077
1.03065
503.4200134
6.1192
1916
493.6499939
5.60529995
26.5
19.0166
5.5153
42.40000153
1.769799948
6.891699791
3.672800064
18.91553099
0.739535
0.000748335
7.933000088
43.40000153
220.981
(1 of ours is worth fewer of theirs.)
(1 of ours is worth more of theirs.)
(1 of ours is worth the same as theirs.)
ISK
INR
IRR
IQD
ILS
JPY
KES
KWD
LBP
LRD
LTL
MKD
MXN
NZD
KPW
OMR
PKR
PHP
XPT
QAR
RUB
SLL
XAG
KRW
SEK
CHF
SYP
TRY
UYU
VND
ZMW
Icelandic Krona
Indian Rupee
Iranian Rial
Iraqi Dinar
Israeli Shekel
Japanese Yen
Kenyan Shilling
Kuwaiti Dinar
Lebanese Pound
Liberian Dollar
Lithuanian Litas
Macedonian Denar
Mexican Peso
New Zealand Dollar
North Korean Won
Omani Rial
Pakistani Rupee
Philippine Peso
Platinum Ounce
Qatari Riyal
Russian Ruble
Sierra Leonean Leone
Silver Ounce
South Korean Won
Swedish Krona
Swiss Franc
Syrian Pound
Turkish Lira
Uruguayan Peso
Vietnamese Dong
Zambian Kwacha
121.15
62.5025
24791
1163.4
3.5567
98.245
86.55
0.2828
1510
78.2
2.55347
45.5
13.142
1.20766
127.17
0.385
105.55
43.37
0.0007
3.6403
32.335
4280
0.04588
1075.08
6.4351
0.90588
112.95
2.0306
21.65
21112
5.31
Geometry notes: Dilations
7
1. Do the following conversions.
a) How many Euro’s is the same as $1000 US Dollars?
b) How many USD is the same as $512 Canadian
Dollars?
c) If you have 25,000 Russian Rubles, how much USD
would that be?
d) If you have 1,000,000 Iranian Rial, how much USD
would that be?
e) If you have $4,500 USD, how much Swedish Krona
would that be?
f) If you have $100 Burmese Kyat, how much USD
would that be ?
g) If you have 125,833 Algerian Dinar, what would
that be in Armenian Dram?
h) If you had 1,400 Gold Ounces, how much USD
would you have?
2. Fill in the lists
STRONGEST CURRENCIES
WEAKEST CURRENCIES
1.
1.
2.
2.
3.
3.
Why do you think those three currencies are so
strong?
Why do you think those three currencies are so
weak?
Geometry notes: Dilations
8
Now move onto dilations
The properties and characteristics of dilations.
Dilation is the term that we use quite regularly in the English language to describe the
enlarging or shrinking of our pupils. Pupils dilate either larger or smaller depending on
the amount of light that enters the eye. This real world example helps us to
understand the use of dilation in geometry as well – dilation is a transformation that
produces an image that is the same shape as the pre-image but is a different size, either larger or smaller.
When we make things bigger using dilation we refer to that as an expansion or enlargement whereas if we use
a dilation to make something smaller we use the term contraction or reduction.
DILATION – CONTRACTION (REDUCTION)
PRE-IMAGE
IMAGE
DILATION – EXPANSION (ENLARGEMENT)
PRE-IMAGE
IMAGE
When the dilation is an enlargement, the scale factor is greater than 1. What this means is that if you have a
scale factor of 3, or sometimes written as 3:1 (image:pre-image), the image is three times bigger
proportionally to the pre-image.
When the dilation is a reduction, the scale factor is between 0 and 1. What this means is that if you have a
scale factor of ½, or sometimes written as 1:2 (image:pre-image), that the image is half the size proportionally
to the pre-image.
So what happens if the scale factor is 1? Nothing!! To multiply anything by 1 maintains the pre-image. It is
like rotating a shape 360 - the shape is not altered in any way – it is an identity transformation.
Q Circle whether the following situations are REDUCTIONS OR ENLARGEMENTS.
a) Scale Factor of 7:1
(image : pre-image)
Reduction
or
Enlargement
b)
DO ,3 ( H )  H '
B
O
B'
c)
Reduction
or
Enlargement
Reduction
or
Enlargement
Geometry notes: Dilations
9
DO ,1.75 ( A)  A '
d)
Reduction
or
e) Scale Factor of 2:3
(image : pre-image)
Enlargement
g)
Reduction
or
f)
Enlargement
D
O,
or
Enlargement
i)
C
C'
C'
B
O
D'
D'
D
B'
D
O
B'
B'
B
O
B
or
Enlargement
J)
Reduction
or
Enlargement
Reduction
k)
F
D'
C
F'
D
E'
D
O
O
B
B'
B'
or
Enlargement
C'
E
D'
C
or
l)
C'
Reduction
(G )  G '
Reduction
h)
C
Reduction
5
3
Enlargement
Reduction
or
Enlargement
O
B
Reduction
or
Enlargement
So what actually happens when a shape is dilated?
B'
The length of each side of the image is equal to the length of the
corresponding side of the pre-image multiplied by the scale factor,
A’B’ = k  AB, B’C’ = k  BC and A’C’ = k  AC.
This dilation has a scale factor of 3:1 or 3.
A’B’:AB
12: 4
3:1
B’C’:BC
16.5: 5.5
3:1
A’C’: AC
15:3
3:1
12 cm
4 cm
A
A'
B
16.5 cm
5.5 cm
5 cm
15 cm
C
C'
Geometry notes: Dilations
10
The distance from the center of the dilation to each point of the image is
equal to the distance from the center of the dilation to each
corresponding point of the pre-image figure times the scale factor,
OB’ = k  OB and OC’ = k  OC.
Scale Factor 1:k
ky
OB = y
OB’ = ky
ky:y
k:1
OC = x
OC’ = kx
kx:x
k:1
B
y
Scale Factor of k:1
B'
C
O
C'
x
kx
Q Answer the following questions about the dilation, centered at O.
a) Is this an enlargement or a reduction?
__________________
Explain how you determined your answer.
A'
A
b) What scale factor do you think this is?
____________
Explain how you determined your answer.
O
B
B'
c) What angle is the same size as OBA?
Explain how you determined your answer.
____________
Q Answer the following questions about the dilation centered at O with a scale factor of 3.
OA = 3, OB = 5 and AB = 4
a) A’B’ = _____________
A'
b) OB’ = _____________
A
c) OA’ = _____________
d) AA’ = _____________ (be careful)
3
O
4
5
B
e) BB’ = _____________ (be careful)
B'
f) What is the ratio of OA:AA’? ______________
Geometry notes: Dilations
11
Activity
Dilating Pictures
How do I do it?
Pick a number of key locations along the edge of the shape and then dilate those points by the correct scale
factor. A compass can make this quite easy. Measure the distance to the pre-image point from the center of
dilation and then mark the remaining lengths on the ray. So for example in the plant example below measure
using your compass to a point on the pre-image and then mark two more of those lengths along the ray to get
three times the pre-image length. Notice to create a scale factor of three I needed a total of three equal
measure the length of the pre-image distance (the pre-image distance from the center and then two more). I
have done a few points to get you started. Do enough points to finish the basic outline of the shape and then
‘rough’ in the rest. Color the final scaled copy.
Geometry notes: Dilations
Pick one of these two to do – (Color and detail the image).
THE ANGRY SHARK (Scale Factor = 2)
Cartoon Female Adventurer (Scale Factor = 2)
12
Geometry notes: Dilations
13
DEFINITION
A dilation with center O and a scale factor of k is a transformation that maps every point P in the plane to
point P’ so that the following properties are true.
NOTATION
DO ,k ( x, y )  (kx, ky )
O is the center of dilation.
k is the value of the scale factor.
PROPERTIES
DILATION PROPERTIES - A dilation is NOT an isometric transformation so its properties differ from the
ones we saw with reflection, rotation and translation. The following properties are preserved between the
pre-image and its image when dilating:




Angle measure (angles stay the same)
Parallelism (things that were parallel are still parallel)
Collinearity (points on a line, remain on the line)
Distance IS NOT preserved!!!
After a dilation, the pre-image and image have the same shape but not the same size.
TRANSFORMATION PROPERTIES – The following properties are
present in dilation:

DISTANCES ARE DIFFERENT (PROPORTIONAL) – The distance
points move during dilation depend on their distance from the
center of dilation - points closer to the center of dilation will
move a shorter distance than those farther away. In our
example PP '  BB '  CC ' and point B is farther away from the
P'
P
B'
B
O
C
C'
center of dilation O than point P, thus BB '  PP ' .


ORIENTATION IS THE SAME – The orientation of the shape is maintained.
SPECIAL POINTS – The center of dilation is an invariant point and does not move in a dilation. If the
pre-image (P) = image (P’) after a dilation then point P was the center of dilation.
Geometry notes: Dilations
14
Verify experimentally the properties of dilations given by a center and a scale factor: the
dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Scale Factor 0 < k < 1
Scale Factor k > 1
B'
B
B'
A'
B
A'
A
A
C
C'
C
C'
B 'C '
k 1
BC
A 'C '
k 1
A’C’ < AC and
AC
A' B '
k 1
A’B’ < AB and
AB
B 'C '
k 1
BC
A 'C '
k 1
A’C’ > AC and
AC
A' B '
k 1
A’B’ > AB and
AB
B’C’ < BC and
B’C’ > BC and
Examples using the coordinate rule of dilation when the center of dilation is the origin.
Example #1 A dilation of 2 with center of dilation O, the origin.
DO ,2 ( x, y )  (2 x, 2 y )
DO ,2 C (1,3)  (2(1), 2(3))  C '(2, 6)
C' (2,6)
C (1,3)
O (0,0)
C (1,3)
O (0,0)
Geometry notes: Dilations
15
Example #2 A dilation of ½ with center of dilation O, the origin.
D
1
O,
2
1 1
(
x
,
y
)

(
x, y )
1
O,
2 2
2
D
D
O,
1
2
1
1
A(6, 4)  ( (6), ( 4))  A '(3, 2)
2
2
1
1
B(2, 8)  ( (2), (8))  B '(1, 4)
2
2
O (0,0)
O (0,0)
A (6,-4)
A' (3,-2)
A (6,-4)
B' (1,-4)
B (2,-8)
Do Worksheets:



Dilation and Lines Practice
Dilation Practice
Thinking about Dilations
B (2,-8)