Dilations & Similarity — Challenge
A dilation multiplies every distance from one fixed point by the same number. Angles survive; lengths do not.
What these problems 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
The point (2,3) is dilated by a scale factor of 3 about the origin. What are its new coordinates?
- (6,9)
- (5,6)
- (2,9)
- (6,3)
- (6,6)
Worked answer
(6,9)
A dilation about the origin multiplies both coordinates by the scale factor.
2· 3 = 6 and 3· 3 = 9, so the image is (6,9).
Adding 3 to each coordinate would be a translation, giving (5,6).
Problem 2
A segment of length 10 is dilated by a scale factor of 12. What is the length of the image?
- 2.5
- 5
- 10
- 15
- 20
Worked answer
5
A dilation multiplies every length by the scale factor.
10·12 = 5.
A factor below 1 shrinks the figure, which is why the answer is less than 10.
Problem 3
A figure is dilated by a scale factor of 4. By what factor is its area multiplied?
- 2
- 4
- 16
- 64
- 8
Worked answer
16
Every length is multiplied by 4.
Area is built from two lengths, so it is multiplied by 4· 4 = 16.
Volume would be multiplied by 43 = 64, which is why that value appears among the options.
Problem 4
A triangle has an area of 5. It is dilated by a scale factor of 3. What is the area of the image?
- 15
- 30
- 45
- 135
- 5
Worked answer
45
Area scales by the square of the scale factor: 32 = 9.
5· 9 = 45 square units.
Multiplying the area by 3 to get 15 scales only one dimension.
Problem 5
What is the result of a dilation with scale factor 1?
- The same figure
- A larger figure
- A smaller figure
- A reflection
- A rotation
Worked answer
The same figure
Each length is multiplied by 1, which leaves it unchanged.
Every point stays where it was, so the image is identical to the original.
A factor above 1 enlarges and a factor between 0 and 1 shrinks; 1 is the dividing case.
Problem 6
The point (-4,6) is dilated by a scale factor of 12 about the origin. What are its new coordinates?
- (-8,12)
- (-2,6)
- (-2,3)
- (-4,3)
- (2,-3)
Worked answer
(-2,3)
Multiply both coordinates by 12.
-4·12 = -2 and 6·12 = 3, giving (-2,3).
The negative sign is unaffected by the dilation, since the factor is positive.
Problem 7
A solid is dilated by a scale factor of 2. By what factor is its volume multiplied?
- 2
- 4
- 6
- 8
- 16
Worked answer
8
Each length doubles.
Volume is built from three lengths, so it is multiplied by 2· 2· 2 = 8.
Area would only be multiplied by 4, which is the usual confusion.
Problem 8
A dilation has a scale factor between 0 and 1. What does it produce?
- A larger figure
- A smaller figure
- A congruent figure
- A rotated figure
- A reflected figure
Worked answer
A smaller figure
Multiplying a length by a number less than 1 makes it shorter.
Every length shrinks by the same proportion, so the image is a smaller copy of the same shape.
It is similar to the original but not congruent, because the sizes differ.