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Dilations
Centered at the Origin
1
KEY
CONCEPTS
• The image and pre-image of a dilated figure are ________________ . The
angles are _____________, but the _____________ will change in size.
• Similar means the figure has the same shape and is ________________ in
size.
• A dilation has a center C and a scale factor k.
• This is a type of transformation that is NOT _________ and NOT
________________ .
• A dilation is a _________________ if 0<k<1 (fraction/decimal less than 1).
• A dilation is an _________________ if k>1.
• If k = 1, then the figures are _________________ .
• FORMULA: Pre-Image x Scale Factor = Image
– (x, y) → (kx, ky)
2
KEY
CONCEPTS
• The image and pre-image of a dilated figure are ________________ . The
angles are _____________, but the _____________ will change in size.
• Similar means the figure has the same shape and is ________________ in
size.
• A dilation has a center C and a scale factor k.
• This is a type of transformation that is NOT _________ and NOT
________________ .
• A dilation is a _________________ if 0<k<1 (fraction/decimal less than 1).
• A dilation is an _________________ if k>1.
• If k = 1, then the figures are _________________ .
• FORMULA: Pre-Image x Scale Factor = Image
– (x, y) → (kx, ky)
3
congruent
similar
sides
proportional
rigid
isometric
reduction
enlargement
congruent
Dilations in the Coordinate Plane
4
Guided Practice
5
Guided Practice
6
3A. Multiply each coordinate by 1/3: Q’(0, 2), R’ (-2, -1), S’(2, -1)
3B. Multiply each coordinate by 2: A’ (4, 2), B’ (0, 6), C’ (-2, 4)
WARM-UP
7
Write true or false for each statement. If false, rewrite the statement to make it true.
________ 1) The dilation of a line segment by a scale factor of greater than 1 results in a shorter image.
________ 2) Similar figures have corresponding sides that are congruent.
________ 3) All regular pentagons are similar to one another.
WARM-UP: Dilations
8
Write true or false for each statement. If false, rewrite the statement to make it true.
________ 1) The dilation of a line segment by a scale factor of greater than 1 results in a shorter image.
________ 2) Similar figures have corresponding sides that are congruent.
________ 3) All regular pentagons are similar to one another.
longer
FALSE
TRUE
FALSE
similar
Determining the Scale Factor
of Similar Figures
9
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞
10
Finding Scale Factor (k)
EX 1: EX 2:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
EX 3:
11
Finding Scale Factor (k)
EX 1: EX 2:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
EX 3:
𝑘 =𝑖𝑚𝑎𝑔𝑒
𝑝𝑟𝑒 − 𝑖𝑚𝑎𝑔𝑒=25
10=5
2𝑘 =
𝑖𝑚𝑎𝑔𝑒
𝑝𝑟𝑒 − 𝑖𝑚𝑎𝑔𝑒=24
30=4
5
𝑘 =𝑖𝑚𝑎𝑔𝑒
𝑝𝑟𝑒 − 𝑖𝑚𝑎𝑔𝑒=48
40=6
5
1) Find the length of two corresponding sides on the two figures.
- Choose a horizontal or a vertical side.
- Count the number of units between the vertices.
2) Divide the image side by the pre-image side.
3) Simplify.
12
Finding Scale Factor (k) of Figures on the Coordinate Plane
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
13
EX 4:
Find the Scale Factor:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
14
EX 4:
Find the Scale Factor:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝒌 =𝑩′𝑪′
𝑩𝑪=𝟑
𝟔=𝟏
𝟐
15
EX 5:
Find the Scale Factor:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
16
EX 5:
Find the Scale Factor:
𝐤 =𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐬𝐢𝐝𝐞 𝐥𝐞𝐧𝐠𝐭𝐡
𝒌 =𝑷′𝑻′
𝑷𝑻=𝟏. 𝟓
𝟑=𝟏
𝟐
17
Find the Scale Factor:
EX 6:
HINT: The ratio of the x- and y-coordinates of the corresponding values of the vertices of an image to the vertices of the pre-image is the same as the scale factor of the dilation.
18
Find the Scale Factor:
EX 6:
HINT: The ratio of the x- and y-coordinates of the corresponding values of the vertices of an image to the vertices of the pre-image is the same as the scale factor of the dilation.
𝒌 =𝒙−𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝑩′
𝒙−𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝑩=
𝟒
𝟐= 2
Error Analysis
19
Vicente says that ΔGHJ could be transformed to ΔG’H’J’ by Dilating it from the origin by a scale factor of 2/3. Is Vicente correct? Why or why not? Explain.
Error Analysis
20
Vicente says that ΔGHJ could be transformed to ΔG’H’J’ by Dilating it from the origin by a scale factor of 2/3. Is Vicente correct? Why or why not? Explain.
𝒌 =𝒙 − 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝑮′
𝒙 − 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝑮=−𝟔
−𝟒=𝟑
𝟐
No, Vicente is not correct. The scale factor is 3/2. The dilation is an enlargement.
21
EX 7: EX 8:Is TVRS a dilation of ABCD? Is ΔMNZ a dilation of ΔXYZ?
22
EX 7: EX 8:
Is TVRS a dilation of ABCD? Is ΔMNZ a dilation of ΔXYZ?
𝒌 =𝑽𝑹
𝑨𝑫=𝟏𝟎
𝟏𝟓=𝟐
𝟑
𝒌 =𝑺𝑹
𝑩𝑨=
𝟖
𝟏𝟎=𝟒
𝟓
Yes, the scale factor is the same for all corresponding sides.
𝒌 =𝑴𝒁
𝑿𝒁=
𝟖
𝟒 + 𝟖=
𝟖
𝟏𝟐=𝟐
𝟑
𝒌 =𝑵𝒁
𝒀𝒁=
𝟔
𝟑 + 𝟔=𝟔
𝟗=𝟐
𝟑
𝒌 =𝑴𝑵
𝑿𝒀=𝟏𝟎
𝟏𝟓=𝟐
𝟑No, the scale factor is NOT the same for all corresponding sides.
Segment P’Q’ is a dilation of segment PQ. The endpoints of each segment are given. Find the scale factor, and classify it as an
enlargement or as a reduction.
• REMEMBER that you can use a _______ of the ___________ or y-coordinates to determine scale factor.
WARM-UP
1) P(0, -10) and Q(5, -15)P’(0, -6) and Q’(3, -9)
3) P(-1, 2) and Q(3, -3)P’(-3, 6) and Q’(9, -9)
2) P(-3, -9) and Q(6, -3)P’(-4, -12) and Q’(8, -4)
Segment P’Q’ is a dilation of segment PQ. The endpoints of each segment are given. Find the scale factor, and classify it as an
enlargement or as a reduction.
• REMEMBER that you can use a _______ of the ___________ or y-coordinates to determine scale factor.
WARM-UP
1) P(0, -10) and Q(5, -15)P’(0, -6) and Q’(3, -9)
Segment P’Q’ is a dilation of segment PQ. The endpoints of each segment are given. Find the scale factor, and classify it as an
enlargement or as a reduction.
• REMEMBER that you can use a _______ of the ___________ or y-coordinates to determine scale factor.
WARM-UP
2) P(-3, -9) and Q(6, -3)P’(-4, -12) and Q’(8, -4)
Segment P’Q’ is a dilation of segment PQ. The endpoints of each segment are given. Find the scale factor, and classify it as an
enlargement or as a reduction.
• REMEMBER that you can use a _______ of the ___________ or y-coordinates to determine scale factor.
WARM-UP
3) P(-1, 2) and Q(3, -3)P’(-3, 6) and Q’(9, -9)
Solving Proportions
What is a Proportion?
• A proportion is an ____________ stating that two ratios are equivalent.
• In a proportion, the ________ ___________
are equal.
EX: 𝟔
𝟖=
𝟑
𝟒
Solving Proportions
• STEPS:
1) Cross-Multiply
2) Divide
EX 1: 𝟕
𝒙=
𝟑
𝟗EX 2:
𝒂
𝟒=
𝟗
𝟏𝟎EX 3:
𝒌
𝟏𝟐.𝟑=
𝟐
𝟑
Solving Proportions
• STEPS:
1) Cross-Multiply
2) Divide
EX 1: 𝟕
𝒙=
𝟑
𝟗EX 2:
𝒂
𝟒=
𝟗
𝟏𝟎EX 3:
𝒌
𝟏𝟐.𝟑=
𝟐
𝟑
Applications of Dilations
Using Scale Factor
EX 1: A photo is 8 in wide by 10 in long. You want to make a reduced color copy of the photo that is 5 in wide for your scrapbook. What is the scale factor you should choose on the copy machine?
A) 50% B) 62.5% C) 160% D) 200%
EX 1: A photo is 8 in wide by 10 in long. You want to make a reduced color copy of the photo that is 5 in wide for your scrapbook. What is the scale factor you should choose on the copy machine?
A) 50% B) 62.5% C) 160% D) 200%
EX 2: Arturo designed a logo for his high school. The logo which is 5 in wide and 8 in long, will be enlarged
and used on a school sweatshirt. On the sweatshirt, the logo will be 12.5 in wide.
A) What is the scale factor for this enlargement?
B) How long will the logo be on the sweatshirt?
EX 2: Arturo designed a logo for his high school. The logo which is 5 in wide and 8 in long, will be enlarged
and used on a school sweatshirt. On the sweatshirt, the logo will be 12.5 in wide.
A) What is the scale factor for this enlargement?
B) How long will the logo be on the sweatshirt?
EX 3: Carla’s optometrist uses medicine to dilate her pupils by a factor of 5/3. The diagram shows
the diameter of Carla’s pupil before dilation. Find the new diameter
once her pupil is dilated.
EX 3: Carla’s optometrist uses medicine to dilate her pupils by a factor of 5/3. The diagram shows
the diameter of Carla’s pupil before dilation. Find the new diameter
once her pupil is dilated.
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