Identify dilations Identify scale factors and solve proportions Draw compositions of transformations

1. Dilation, Scale Factor, & Proportion
The student is able to (I can):
• Identify dilations
• Identify scale factors and use scale factors to solve• Identify scale factors and use scale factors to solve
problems
• Write and simplify ratios
• Use proportions to solve problems
• Draw compositions of transformations

2. dilationdilationdilationdilation – a transformation that changes the size of a figure
but not the shape.
Example:
Tell whether each transformation appears to be a dilation.
1. 2.
SS
yes no

3. ratioratioratioratio – a comparison of two numbers by division.
The ratio of two numbers a and b, where b does not equal 0
(b ≠ 0) can be written as
a to b
a : b
a
b
Example: The ratio comparing 1 and 2 can be written 1 to 2,
1 : 2, or .
To compare more than two numbers, use “dot”
notation. Ex. 3 : 7 : 9
b
1
2

4. proportionproportionproportionproportion – an equation stating that two ratios are equal.
Two sets of numbers are proportionalproportionalproportionalproportional if they use the same
ratio.
Example: or a : b = c : d
a c
b d
Cross Products Property
In a proportion, if , and b and d ≠ 0, then ad = bc
a c
b d


5. 


X´´´´
Y´´´´
Z´´´´



P
X
Y
Z
center of
dilation
scalescalescalescale factorfactorfactorfactor – the ratio of the image to the preimage.
If k < 1, the figure gets smaller; if k > 1, the figure gets larger.
X Y Y Z X Z
k
XY YZ XZ
     
  

6. Examples
1. What is the scale factor of the dilation?
10
24
5
12
2. If you are enlarging a 4x6 photo by a scale factor of 4,
what are the new dimensions?

7. Examples
1. What is the scale factor of the dilation?
10
24
5
12
5 1 12 1
(or )
10 2 24 2
k k   
2. If you are enlarging a 4x6 photo by a scale factor of 4,
what are the new dimensions?
4(3) = 12 6(3) = 18
New dimensions = 12x18
10 2 24 2

8. Examples Solve each proportion:
1.
2.
3
8 32
x

4 2
5x

3.
2
6 3
x x 


9. Examples Solve each proportion:
1.
8x = 96 x = 12
2.
2x = 20 x = 10
3
8 32
x

4 2
5x

2x = 20 x = 10
3.
3x = 6(x – 2)
3x = 6x – 12
–3x = –12 x = 4
2
6 3
x x 


10. Examples 4. The ratio of the angles of a triangle is
2: 2: 5. What is the measure of each
angle?

11. Examples 4. The ratio of the angles of a triangle is
2: 2: 5. What is the measure of each
angle?
2x + 2x + 5x = 180˚
9x = 180˚
x = 20
220 = 40˚
220 = 40˚
520 = 100˚

12. composition ofcomposition ofcomposition ofcomposition of transformationstransformationstransformationstransformations – performing two or more
transformations sequentially (one after another) to a
figure.
An example of a composition is a glideglideglideglide
reflectionreflectionreflectionreflection: we reflect the figure and then
translate it along a vector.

13. With most compositions, it is important to perform them in
the order given.
ReflectReflectReflectReflect across theacross theacross theacross the
xxxx----axisaxisaxisaxis and then rotaterotaterotaterotate
90909090.
RotateRotateRotateRotate 90909090 and then
reflect across thereflect across thereflect across thereflect across the
xxxx----axisaxisaxisaxis.

14. Examples
Graph the transformations
1. (1, 4), (–2, 1), (–4, 1): Translate along the vector 〈–3, 1〉
and then reflect across the y-axis
2. (2, 1), (3, 5), (5, 2): Rotate 180° about the origin and then
reflect across the line y = 2

15. Examples
Graph the transformations
1. (1, 4), (–2, 1), (–4, 1): Translate along the vector 〈–3, 1〉
and then reflect across the y-axis

16. Examples
Graph the transformations
2. (2, 1), (3, 5), (5, 2): Rotate 180° about the origin and then
reflect across the line y = 2