Dilations & Similarity — Print
A dilation multiplies every distance from one fixed point by the same number. Angles survive; lengths do not.
What the practice 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.
A projector throws a small slide onto a big wall. The picture on the wall is larger. Is it a different picture?
The hill is still a hill. The sun is still round. Nothing got bent.
Move the wall twice as far from the lamp and the picture becomes twice as wide and twice as tall.
Every length doubled. But the corner where the roof meets the wall is still the same corner — angles did not change at all.
That is the difference between a dilation and every move in the last unit: distances change, angles do not.
So a dilation gives you a shape that is the same shape at a different size. Mathematicians call that similar, and the multiplier — 2 here — is the scale factor k.
What a dilation does
Pick a centre. For every point of the shape, measure its distance from the centre and multiply that distance by k, staying on the same ray.
With the centre at the origin the rule is beautifully simple: (x, y) → (kx, ky).
The image is a scaled copy — every length multiplied by k, every angle exactly as it was.
Angles are untouched
A dilation stretches distances but never bends anything. That is why the image is the same shape, and why similar figures have equal angles.
Reading the scale factor
If the image is bigger, k > 1. If smaller, k is a fraction between 0 and 1. k = 1 leaves everything alone.
To find k from a picture, divide any image length by the matching pre-image length: k = imageoriginal.
Do it with two different pairs of sides. If you get the same k both times, the figures really are similar.
Which goes on top?
New over old. If a side of 4 becomes 10, then k = 104 = 2.5 — bigger than one, and the picture got bigger. Sanity check it that way every time.
Similar versus congruent
Congruent means identical. Similar means the same shape at possibly a different scale.
Every congruent pair is similar, with k = 1. Not every similar pair is congruent.
In similar figures: angles are equal, and sides are proportional. Those two sentences solve almost every similarity problem.
Finding a missing side
Set up a proportion with matching sides in matching positions, then cross-multiply.
If △ ABC ∼ △ DEF then ABDE = BCEF = ACDF.
Keep the same triangle on top in every fraction. Flipping one of them midway is the classic slip.
Perimeter scales by k, area by k2
Double every side and the perimeter doubles, but the area becomes four times as large. For volume it is k3.
Worked examples
Dilating from the origin
Dilate A(6, -4) by a scale factor of 12, centred at the origin.
- Rule: (x, y) → (12x, 12y) Centre at the origin makes it pure multiplication.
- 12(6) = 3 and 12(-4) = -2 Halve both coordinates, signs and all.
- Sanity check: k < 1, so the point moved closer to the origin. (3, -2) is indeed nearer than (6, -4).
A'(3, -2)
Finding the scale factor
A rectangle 8 by 12 is dilated into a rectangle 20 by 30. Find k.
- Use a matching pair: k = 208 = 2.5 New over old.
- Check with the other pair: 3012 = 2.5 ✓ Both give the same factor, so it really is a dilation.
k = 2.5
Missing side in similar triangles
△ ABC ∼ △ DEF with AB = 6, DE = 9 and BC = 8. Find EF.
- Scale factor: k = DEAB = 96 = 32 Going from △ ABC to △ DEF.
- EF = k · BC = 32 × 8 Every side in △ DEF is 32 times its partner.
- EF = 12 Check: 128 = 1.5, matching 96.
EF = 12
Words to know
- Dilation
- A resizing from a centre point by a scale factor k.
- Centre of dilation
- The one point that does not move.
- Scale factor
- The multiplier k. k > 1 enlarges, 0 < k < 1 shrinks.
- Similar
- Same shape, possibly different size. Written △ ABC ∼ △ DEF.
Mistakes to watch for
Dividing old by new. The scale factor is newold, so a side of 4 becoming 10 gives k = 2.5. Get it upside down and you get 0.4, which claims the picture shrank when you can see that it grew.
Adding the scale factor instead of multiplying. A dilation multiplies every distance from the centre. Adding 2 to each side turns a 3-4-5 triangle into 5-6-7, which is not even the same shape any more.
Assuming area scales by k too. Double every side and you get four times the area, because area is a length times a length and both of them doubled. For volume it is k3, for the same reason.
Flipping one fraction in a proportion. ABDE = BCEF keeps the first triangle on top in both. Turn one over and you are solving a different, wrong equation that often still gives a tidy-looking answer.
Thales of Miletus measured the height of the Great Pyramid around 600 BC using nothing but its shadow and his own.
He waited until his shadow was exactly as long as he was tall, then measured the pyramid's shadow. Same scale factor, so that length was the pyramid's height.
It is said the Egyptian priests were astonished. The trick is in this unit: two similar triangles, one of them enormous.
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