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Amy Ferguson Moncure

7th Grade Math · Unit 6

Dilations

A dilation resizes a figure from the origin. Multiply every coordinate by the scale factor.

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Key Ideas Easy Medium Hard
What to Remember

Center at the origin

Every dilation on this page has its center at the origin.

Multiply both coordinates by the scale factor. The image of (x, y) is that factor times x and that factor times y.

A dilation is a resize

A dilation is a resize, the same idea as a scale drawing or a graphic that is enlarged or reduced.

Enlarge or reduce

A scale factor greater than 1 moves the image farther from the origin.

A scale factor less than 1 moves the image closer to the origin. The shape stays the same.

Factors on this page

The scale factor is 2, 3, 4, one half, or one fourth.

For one half, start with even coordinates. For one fourth, start with multiples of 4 so the image lands on whole-number points.

Level 1

Easy

The scale factor is 2 or 3. The points are in the first quadrant.

Example
Dilate ABC by a scale factor of 2 with center at the origin. The original points are A(1, 1), B(3, 1), C(2, 3).

(x, y) becomes (2x, 2y).

A(1, 1) becomes A'(2, 2).

B(3, 1) becomes B'(6, 2).

C(2, 3) becomes C'(4, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(2, 2), B'(6, 2), C'(4, 6)

1
Dilate ABC by a scale factor of 2 with center at the origin. The original points are A(1, 2), B(2, 2), C(1, 4).
Show Answer

(x, y) becomes (2x, 2y).

A(1, 2) becomes A'(2, 4).

B(2, 2) becomes B'(4, 4).

C(1, 4) becomes C'(2, 8).

−112345−1123456789ABCA'B'C'

Answer: A'(2, 4), B'(4, 4), C'(2, 8)

2
Dilate ABC by a scale factor of 3 with center at the origin. The original points are A(1, 1), B(2, 1), C(1, 2).
Show Answer

(x, y) becomes (3x, 3y).

A(1, 1) becomes A'(3, 3).

B(2, 1) becomes B'(6, 3).

C(1, 2) becomes C'(3, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(3, 3), B'(6, 3), C'(3, 6)

3
Dilate ABCD by a scale factor of 2 with center at the origin. The original points are A(2, 1), B(4, 1), C(3, 2), D(2, 3).
Show Answer

(x, y) becomes (2x, 2y).

A(2, 1) becomes A'(4, 2).

B(4, 1) becomes B'(8, 2).

C(3, 2) becomes C'(6, 4).

D(2, 3) becomes D'(4, 6).

−1123456789−11234567ABCDA'B'C'D'

Answer: A'(4, 2), B'(8, 2), C'(6, 4), D'(4, 6)

4
Dilate ABC by a scale factor of 3 with center at the origin. The original points are A(1, 1), B(2, 1), C(2, 2).
Show Answer

(x, y) becomes (3x, 3y).

A(1, 1) becomes A'(3, 3).

B(2, 1) becomes B'(6, 3).

C(2, 2) becomes C'(6, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(3, 3), B'(6, 3), C'(6, 6)

5
Dilate ABC by a scale factor of 2 with center at the origin. The original points are A(1, 1), B(2, 3), C(3, 1).
Show Answer

(x, y) becomes (2x, 2y).

A(1, 1) becomes A'(2, 2).

B(2, 3) becomes B'(4, 6).

C(3, 1) becomes C'(6, 2).

−11234567−11234567ABCA'B'C'

Answer: A'(2, 2), B'(4, 6), C'(6, 2)

6
Dilate ABC by a scale factor of 3 with center at the origin. The original points are A(1, 1), B(1, 2), C(2, 2).
Show Answer

(x, y) becomes (3x, 3y).

A(1, 1) becomes A'(3, 3).

B(1, 2) becomes B'(3, 6).

C(2, 2) becomes C'(6, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(3, 3), B'(3, 6), C'(6, 6)

7
Dilate ABC by a scale factor of 2 with center at the origin. The original points are A(1, 3), B(3, 3), C(2, 1).
Show Answer

(x, y) becomes (2x, 2y).

A(1, 3) becomes A'(2, 6).

B(3, 3) becomes B'(6, 6).

C(2, 1) becomes C'(4, 2).

−11234567−11234567ABCA'B'C'

Answer: A'(2, 6), B'(6, 6), C'(4, 2)

8
Dilate ABC by a scale factor of 3 with center at the origin. The original points are A(1, 2), B(2, 1), C(2, 2).
Show Answer

(x, y) becomes (3x, 3y).

A(1, 2) becomes A'(3, 6).

B(2, 1) becomes B'(6, 3).

C(2, 2) becomes C'(6, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(3, 6), B'(6, 3), C'(6, 6)

9
Dilate ABCD by a scale factor of 2 with center at the origin. The original points are A(1, 1), B(3, 1), C(3, 2), D(1, 2).
Show Answer

(x, y) becomes (2x, 2y).

A(1, 1) becomes A'(2, 2).

B(3, 1) becomes B'(6, 2).

C(3, 2) becomes C'(6, 4).

D(1, 2) becomes D'(2, 4).

−11234567−112345ABCDA'B'C'D'

Answer: A'(2, 2), B'(6, 2), C'(6, 4), D'(2, 4)

10
Dilate ABC by a scale factor of 3 with center at the origin. The original points are A(2, 1), B(1, 2), C(2, 2).
Show Answer

(x, y) becomes (3x, 3y).

A(2, 1) becomes A'(6, 3).

B(1, 2) becomes B'(3, 6).

C(2, 2) becomes C'(6, 6).

−11234567−11234567ABCA'B'C'

Answer: A'(6, 3), B'(3, 6), C'(6, 6)

Level 2

Medium

The scale factor is 4 or one half.

Example
Dilate ABC by a scale factor of 4 with center at the origin. The original points are A(1, 1), B(2, 1), C(1, 2).

(x, y) becomes (4x, 4y).

A(1, 1) becomes A'(4, 4).

B(2, 1) becomes B'(8, 4).

C(1, 2) becomes C'(4, 8).

−1123456789−1123456789ABCA'B'C'

Answer: A'(4, 4), B'(8, 4), C'(4, 8)

1
Dilate ABC by a scale factor of 12 with center at the origin. The original points are A(2, 2), B(4, 2), C(2, 6).
Show Answer

(x, y) becomes (12x, 12y).

A(2, 2) becomes A'(1, 1).

B(4, 2) becomes B'(2, 1).

C(2, 6) becomes C'(1, 3).

−112345−11234567ABCA'B'C'

Answer: A'(1, 1), B'(2, 1), C'(1, 3)

2
Dilate ABC by a scale factor of 4 with center at the origin. The original points are A(1, 2), B(2, 2), C(2, 1).
Show Answer

(x, y) becomes (4x, 4y).

A(1, 2) becomes A'(4, 8).

B(2, 2) becomes B'(8, 8).

C(2, 1) becomes C'(8, 4).

−1123456789−1123456789ABCA'B'C'

Answer: A'(4, 8), B'(8, 8), C'(8, 4)

3
Dilate ABC by a scale factor of 12 with center at the origin. The original points are A(2, 4), B(6, 4), C(4, 2).
Show Answer

(x, y) becomes (12x, 12y).

A(2, 4) becomes A'(1, 2).

B(6, 4) becomes B'(3, 2).

C(4, 2) becomes C'(2, 1).

−11234567−112345ABCA'B'C'

Answer: A'(1, 2), B'(3, 2), C'(2, 1)

4
Dilate ABC by a scale factor of 4 with center at the origin. The original points are A(1, 1), B(1, 2), C(2, 2).
Show Answer

(x, y) becomes (4x, 4y).

A(1, 1) becomes A'(4, 4).

B(1, 2) becomes B'(4, 8).

C(2, 2) becomes C'(8, 8).

−1123456789−1123456789ABCA'B'C'

Answer: A'(4, 4), B'(4, 8), C'(8, 8)

5
Dilate ABCD by a scale factor of 12 with center at the origin. The original points are A(4, 2), B(6, 2), C(6, 4), D(4, 6).
Show Answer

(x, y) becomes (12x, 12y).

A(4, 2) becomes A'(2, 1).

B(6, 2) becomes B'(3, 1).

C(6, 4) becomes C'(3, 2).

D(4, 6) becomes D'(2, 3).

−11234567−11234567ABCDA'B'C'D'

Answer: A'(2, 1), B'(3, 1), C'(3, 2), D'(2, 3)

6
Dilate ABC by a scale factor of 4 with center at the origin. The original points are A(2, 1), B(1, 1), C(1, 2).
Show Answer

(x, y) becomes (4x, 4y).

A(2, 1) becomes A'(8, 4).

B(1, 1) becomes B'(4, 4).

C(1, 2) becomes C'(4, 8).

−1123456789−1123456789ABCA'B'C'

Answer: A'(8, 4), B'(4, 4), C'(4, 8)

7
Dilate ABC by a scale factor of 12 with center at the origin. The original points are A(6, 2), B(8, 2), C(6, 4).
Show Answer

(x, y) becomes (12x, 12y).

A(6, 2) becomes A'(3, 1).

B(8, 2) becomes B'(4, 1).

C(6, 4) becomes C'(3, 2).

−1123456789−112345ABCA'B'C'

Answer: A'(3, 1), B'(4, 1), C'(3, 2)

8
Dilate ABC by a scale factor of 4 with center at the origin. The original points are A(1, 1), B(2, 2), C(1, 2).
Show Answer

(x, y) becomes (4x, 4y).

A(1, 1) becomes A'(4, 4).

B(2, 2) becomes B'(8, 8).

C(1, 2) becomes C'(4, 8).

−1123456789−1123456789ABCA'B'C'

Answer: A'(4, 4), B'(8, 8), C'(4, 8)

9
Dilate ABC by a scale factor of 12 with center at the origin. The original points are A(2, 2), B(6, 2), C(4, 6).
Show Answer

(x, y) becomes (12x, 12y).

A(2, 2) becomes A'(1, 1).

B(6, 2) becomes B'(3, 1).

C(4, 6) becomes C'(2, 3).

−11234567−11234567ABCA'B'C'

Answer: A'(1, 1), B'(3, 1), C'(2, 3)

10
Dilate ABCD by a scale factor of 4 with center at the origin. The original points are A(1, 1), B(2, 1), C(2, 2), D(1, 2).
Show Answer

(x, y) becomes (4x, 4y).

A(1, 1) becomes A'(4, 4).

B(2, 1) becomes B'(8, 4).

C(2, 2) becomes C'(8, 8).

D(1, 2) becomes D'(4, 8).

−1123456789−1123456789ABCDA'B'C'D'

Answer: A'(4, 4), B'(8, 4), C'(8, 8), D'(4, 8)

Level 3

Hard

The scale factor is one fourth. Some points are outside the first quadrant.

Example
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(4, 4), B(8, 8), C(8, 4).

(x, y) becomes (14x, 14y).

A(4, 4) becomes A'(1, 1).

B(8, 8) becomes B'(2, 2).

C(8, 4) becomes C'(2, 1).

−1123456789−1123456789ABCA'B'C'

Answer: A'(1, 1), B'(2, 2), C'(2, 1)

1
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(4, 4), B(8, 4), C(8, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(4, 4) becomes A'(1, 1).

B(8, 4) becomes B'(2, 1).

C(8, 8) becomes C'(2, 2).

−1123456789−1123456789ABCA'B'C'

Answer: A'(1, 1), B'(2, 1), C'(2, 2)

2
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(4, 8), B(12, 4), C(8, 12).
Show Answer

(x, y) becomes (14x, 14y).

A(4, 8) becomes A'(1, 2).

B(12, 4) becomes B'(3, 1).

C(8, 12) becomes C'(2, 3).

−112345678910111213−112345678910111213ABCA'B'C'

Answer: A'(1, 2), B'(3, 1), C'(2, 3)

3
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(8, 4), B(12, 4), C(8, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(8, 4) becomes A'(2, 1).

B(12, 4) becomes B'(3, 1).

C(8, 8) becomes C'(2, 2).

−112345678910111213−1123456789ABCA'B'C'

Answer: A'(2, 1), B'(3, 1), C'(2, 2)

4
Dilate ABCD by a scale factor of 14 with center at the origin. The original points are A(4, 4), B(12, 4), C(12, 8), D(4, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(4, 4) becomes A'(1, 1).

B(12, 4) becomes B'(3, 1).

C(12, 8) becomes C'(3, 2).

D(4, 8) becomes D'(1, 2).

−112345678910111213−1123456789ABCDA'B'C'D'

Answer: A'(1, 1), B'(3, 1), C'(3, 2), D'(1, 2)

5
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(−4, 4), B(−8, 4), C(−4, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(−4, 4) becomes A'(−1, 1).

B(−8, 4) becomes B'(−2, 1).

C(−4, 8) becomes C'(−1, 2).

−9−8−7−6−5−4−3−2−11−1123456789ABCA'B'C'

Answer: A'(−1, 1), B'(−2, 1), C'(−1, 2)

6
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(4, −4), B(8, −4), C(4, −8).
Show Answer

(x, y) becomes (14x, 14y).

A(4, −4) becomes A'(1, −1).

B(8, −4) becomes B'(2, −1).

C(4, −8) becomes C'(1, −2).

−1123456789−9−8−7−6−5−4−3−2−11ABCA'B'C'

Answer: A'(1, −1), B'(2, −1), C'(1, −2)

7
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(−4, −4), B(−8, −4), C(−8, −8).
Show Answer

(x, y) becomes (14x, 14y).

A(−4, −4) becomes A'(−1, −1).

B(−8, −4) becomes B'(−2, −1).

C(−8, −8) becomes C'(−2, −2).

−9−8−7−6−5−4−3−2−11−9−8−7−6−5−4−3−2−11ABCA'B'C'

Answer: A'(−1, −1), B'(−2, −1), C'(−2, −2)

8
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(−8, 4), B(−4, 8), C(−8, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(−8, 4) becomes A'(−2, 1).

B(−4, 8) becomes B'(−1, 2).

C(−8, 8) becomes C'(−2, 2).

−9−8−7−6−5−4−3−2−11−1123456789ABCA'B'C'

Answer: A'(−2, 1), B'(−1, 2), C'(−2, 2)

9
Dilate ABC by a scale factor of 14 with center at the origin. The original points are A(8, −4), B(4, −8), C(12, −4).
Show Answer

(x, y) becomes (14x, 14y).

A(8, −4) becomes A'(2, −1).

B(4, −8) becomes B'(1, −2).

C(12, −4) becomes C'(3, −1).

−112345678910111213−9−8−7−6−5−4−3−2−11ABCA'B'C'

Answer: A'(2, −1), B'(1, −2), C'(3, −1)

10
Dilate ABCD by a scale factor of 14 with center at the origin. The original points are A(−4, 4), B(4, 4), C(4, 8), D(−4, 8).
Show Answer

(x, y) becomes (14x, 14y).

A(−4, 4) becomes A'(−1, 1).

B(4, 4) becomes B'(1, 1).

C(4, 8) becomes C'(1, 2).

D(−4, 8) becomes D'(−1, 2).

−5−4−3−2−112345−1123456789ABCDA'B'C'D'

Answer: A'(−1, 1), B'(1, 1), C'(1, 2), D'(−1, 2)

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