Dilations & Similarity — Practice
A dilation multiplies every distance from one fixed point by the same number. Angles survive; lengths do not.
What these questions cover
These questions use 14 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Scale factor — The number every length is multiplied by.
- Similar figures — Same shape, different size: equal angles, proportional sides.
- Corresponding sides — Sides in matching positions in two similar figures.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Volume — Space inside, in cubic units.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Scale — What one grid step is worth — and where the axis starts.
- Dilation — A resizing from a centre point by a scale factor k.
- Similar — Same shape, possibly different size. Written △ ABC ∼ △ DEF.
- Corresponding angles — Same position at each crossing. Equal when the lines are parallel.
- Segment — Two endpoints and everything between them. Written AB.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
Dilate A(3, 5) by a factor of 2, centred at the origin.
Worked answer
(6, 10)
3 × 2 = 6
5 × 2 = 10
A'(6, 10)
Problem 2
Dilate B(-8, 12) by a factor of 14, centred at the origin.
Worked answer
(-2, 3)
-8 × 14 = -2
12 × 14 = 3
B'(-2, 3)
Problem 3
A side of length 5 becomes a side of length 15 after a dilation. Find the scale factor.
Worked answer
3
k = 155
k = 3
Problem 4
A side of length 24 becomes 18. Find the scale factor.
Worked answer
0.75
k = 1824
k = 34 = 0.75
Problem 5
A dilation has k = 0.4. The image is
- smaller than the original
- larger than the original
- the same size
- turned upside down
Worked answer
smaller than the original
0.4 < 1
So every distance shrinks
The image is smaller
Problem 6
△ ABC ∼ △ DEF with AB = 4, DE = 10 and BC = 6. Find EF.
Worked answer
15
k = 104 = 2.5
EF = 2.5 × 6
EF = 15
Problem 7
△ PQR ∼ △ STU with PQ = 9, ST = 6 and SU = 8. Find PR.
Worked answer
12
k = 96 = 32
PR = 32 × 8
PR = 12
Problem 8
Dilate the segment from (2, 2) to (6, 4) by a factor of 3 about the origin. Give the two image points.
Worked answer
(6, 6) and (18, 12)
(2,2) → (6,6)
(6,4) → (18,12)
Problem 9
A rectangle has perimeter 20. It is dilated by a factor of 3. Find the new perimeter.
Worked answer
60
Every side triples
So the perimeter triples
20 × 3 = 60
Problem 10
A triangle has area 12 cm2. After a dilation with k = 2, find the new area.
Worked answer
48
k2 = 4
12 × 4 = 48
48 cm2
Problem 11
Which is always true for two similar figures?
- Corresponding angles are equal
- Corresponding sides are equal
- Perimeters are equal
- Areas are equal
Worked answer
Corresponding angles are equal
Angles never change under a dilation
Sides change by the factor k
So only the angle statement is always true
Problem 12
A photograph 4 inches wide is enlarged so it is 10 inches wide. If it was 6 inches tall, how tall is the enlargement?
Worked answer
15
k = 104 = 2.5
6 × 2.5 = 15
15 inches
Problem 13
Two similar triangles have areas 9 cm2 and 81 cm2. Find the scale factor between them.
Worked answer
3
k2 = 819 = 9
k = 3
Problem 14
A dilation centred at (0, 0) maps (6, -9) to (-4, 6). Find k.
Worked answer
-2/3
-46 = -23
Check the other: 6-9 = -23 ✓
k = -23 — a negative factor also turns the figure 180°
Problem 15
A 6-foot person casts a 4-foot shadow. At the same moment a tree casts a 22-foot shadow. How tall is the tree?
Worked answer
33
heightshadow is the same for both
64 = h22
h = 6 × 224 = 33 feet
Problem 16
A cube is dilated by a factor of 2. By what factor does its volume increase?
Worked answer
8
k3 = 23
= 8
A doubled cube holds eight times as much