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Math · Geometry & Proportional Reasoning

Dilations & Scale Factor: Enlarge, Reduce, Coordinates & Similar Figures

Build on rigid transformations by using scale factors to enlarge or reduce lengths and coordinate figures, identify similar figures, and predict how perimeter and area change.

Grades 5–7 · 40 practice questions · Original material

Learning objectiveApply a scale factor to lengths and coordinates, recover a missing scale factor, distinguish dilations from rigid motions, and reason about perimeter and area in similar figures.

The idea

A dilation changes size by multiplying every distance from a centre by the same scale factor. If k>1, the image is enlarged; if 0<k<1, it is reduced.

When the centre is the origin, a coordinate dilation follows (x,y)→(kx,ky). Unlike a translation, reflection or rotation, a dilation does not usually preserve length.

Dilations preserve angle sizes and overall shape, so the original and image are similar. Corresponding lengths share the same scale factor, perimeter scales by k, and area scales by k².

Worked examples

Example 1 · Enlarge a length

A 6 cm segment is dilated by scale factor 1.5. Find the image length.

Multiply the original length by k.
6×1.5=9.
The image is longer because k>1.

9 cm.

Example 2 · Reduce a figure

A rectangle is 10 cm by 6 cm. Dilate it by k=1/2.

Multiply each side by 1/2.
10×1/2=5 and 6×1/2=3.
Angle sizes stay unchanged.

5 cm by 3 cm.

Example 3 · Dilate a coordinate

Dilate (−3,4) from the origin by k=2.

Multiply both coordinates by 2.
−3×2=−6 and 4×2=8.
Use the same factor on every vertex.

(−6, 8).

Example 4 · Find the scale factor

A side changes from 8 cm to 12 cm. What is k?

Use image ÷ original.
12÷8=1.5.
Check: 8×1.5=12.

k=1.5.

Example 5 · Area does not scale linearly

A 3 cm by 5 cm rectangle is dilated by k=2. Compare areas.

Original area: 3×5=15 cm².
New sides: 6 cm and 10 cm, so new area is 60 cm².
60÷15=4=2².

Area is multiplied by 4.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Adding the scale factor instead of multiplying.

Why this fails: A dilation multiplies every distance from the centre by the same factor.

Multiplying only one coordinate.

Why this fails: For a dilation from the origin, both x and y are multiplied by k.

Calling a dilation a rigid transformation.

Why this fails: Rigid motions preserve distance; a dilation normally changes lengths.

Using original ÷ image to find k.

Why this fails: Scale factor is image length ÷ corresponding original length.

Multiplying area by k instead of k².

Why this fails: Both length and width scale by k, so area scales by k×k.

Foundation

Foundation practice

Apply scale factors to one-dimensional lengths.

Enlarge

1. A 4 cm segment is dilated by k=2. Find the image length.

Reduce

2. A 12 cm segment is dilated by k=1/2.

Enlarge

3. A 5 m side is dilated by k=3.

Reduce

4. A 20 mm side is dilated by k=0.25.

Scale factor

5. A length changes from 7 to 14. Find k.

Scale factor

6. A length changes from 18 to 6. Find k.

Classify

7. Does k=1.4 enlarge or reduce?

Classify

8. Does k=0.6 enlarge or reduce?

Identity

9. What happens when k=1?

Reason

10. Can a positive dilation use k=0?

Core

Core practice

Dilate coordinates from the origin and compare figures.

Coordinate

11. Dilate (2,3) by k=2.

Coordinate

12. Dilate (−4,5) by k=1/2.

Coordinate

13. Dilate (6,−2) by k=3.

Coordinate

14. Dilate (−8,−4) by k=1/4.

Origin

15. What happens to (0,0) under a dilation centred at the origin?

Triangle

16. Triangle A(1,1), B(3,1), C(1,4) is dilated by k=2. Find A′.

Triangle

17. Using the same dilation, find B′.

Triangle

18. Using the same dilation, find C′.

Compare

19. A side of the original triangle is 3 units. How long is the corresponding side after k=2?

Angles

20. A 50° angle is in a figure dilated by k=3. What is its image angle?

Reasoning

Reasoning practice

Work backward and distinguish similarity from rigid motion.

Inverse

21. An image length is 18 after k=3. What was the original length?

Inverse coordinate

22. A point becomes (10,−6) after k=2. What was the original?

Scale factor

23. Corresponding sides are 9 and 15, original then image. Find k.

Similarity

24. Two rectangles are 4×7 and 8×14. Are they related by one dilation?

Not similar

25. Rectangles are 4×7 and 8×13. Can one dilation map the first to the second?

Rigid or not

26. Which preserves length: rotation or dilation with k=2?

Composition

27. Dilate by k=2, then by k=1/2 about the same centre. What is the net scale factor?

Composition

28. Dilate by k=3 then k=2 about the same centre. Net scale factor?

Error analysis

29. A student maps (3,4) to (6,4) for k=2. What is wrong?

Rule

30. Which rule is a dilation from the origin by k=1/3?

Stretch

Stretch practice

Connect scale factor to perimeter, area and multi-step reasoning.

Perimeter

31. A triangle has perimeter 18 cm. Dilate by k=2. New perimeter?

Perimeter

32. A rectangle has perimeter 50 cm. Dilate by k=0.4. New perimeter?

Area

33. A shape has area 12 cm². Dilate by k=3. New area?

Area

34. A shape has area 80 cm². Dilate by k=1/2. New area?

Recover k

35. Area changes from 25 cm² to 100 cm² under a positive dilation. Find k.

Compare

36. If k=4, by what factor does perimeter change?

Compare

37. If k=4, by what factor does area change?

Coordinate

38. Triangle vertices are (−1,2),(2,2),(2,5). After k=2, what is the image of (2,5)?

Synthesis

39. Why are a figure and its dilation called similar rather than congruent when k≠1?

Model

40. Why does the rule (x,y)→(2x,2y) preserve slopes of nonvertical segments from point pairs?

Answer key

1. 8 cm
4×2=8.
2. 6 cm
12×1/2=6.
3. 15 m
5×3=15.
4. 5 mm
20×0.25=5.
5. 2
14÷7=2.
6. 1/3
6÷18=1/3.
7. Enlarge.
A scale factor greater than 1 enlarges.
8. Reduce.
A positive scale factor less than 1 reduces.
9. The figure stays the same size and position relative to the centre.
Every distance is multiplied by 1.
10. Not in this lesson’s dilation model.
k=0 would collapse every point to the centre instead of producing a similar figure.
11. (4, 6)
Multiply both coordinates by 2.
12. (−2, 2.5)
Multiply x and y by 1/2.
13. (18, −6)
Multiply both coordinates by 3.
14. (−2, −1)
Multiply both coordinates by 1/4.
15. It stays at (0,0).
The centre of dilation is fixed.
16. (2, 2)
Multiply both coordinates by 2.
17. (6, 2)
(3,1)→(6,2).
18. (2, 8)
(1,4)→(2,8).
19. 6 units
Corresponding lengths scale by k.
20. 50°
Dilations preserve angle measure.
21. 6
Undo the dilation: 18÷3=6.
22. (5, −3)
Divide both image coordinates by 2.
23. 5/3
15÷9=5/3.
24. Yes, k=2.
Both corresponding dimensions have the same ratio.
25. No.
The width ratio is 2 but the height ratio is 13/7.
26. Rotation.
The dilation doubles lengths.
27. 1
Multiply factors: 2×1/2=1.
28. 6
Scale factors multiply.
29. Only x was scaled.
Both coordinates must be multiplied by 2.
30. (x,y)→(x/3,y/3)
Both coordinates receive the same factor.
31. 36 cm
Perimeter scales by k.
32. 20 cm
50×0.4=20.
33. 108 cm²
Area scales by 3²=9; 12×9=108.
34. 20 cm²
Area scales by (1/2)²=1/4.
35. 2
Area factor is 4, so k=√4=2.
36. 4
Every side length is multiplied by 4.
37. 16
Area factor is k²=16.
38. (4, 10)
Multiply both coordinates by 2.
39. They have the same shape and corresponding angles, but different size.
Congruent figures must also have equal corresponding lengths.
40. Both horizontal and vertical changes are multiplied by the same factor.
The ratio Δy/Δx stays unchanged.

Related practice

Coordinate Transformations: Translations, Reflections & Rotations Ratios & Unit Rates: Equivalent Ratios, Tables, Double Number Lines, Compare Coordinate Plane: Ordered Pairs, Quadrants, Reflections & Distance