1.6 Dilations — Transformations

Standards: G-CO.B.6 · G-SRT.A.1

A recipe for four people needs doubling for eight. Every quantity doubles — flour, sugar, water, all of it, by the same factor.

What does not double is the ratios. Twice the flour and twice the water is still the same batter. It is a bigger cake, not a different one.

That is a dilation, and it is the first transformation this unit has met that changes size. Everything before it kept the figure congruent. This one keeps it only similar.

If a 6-inch cake becomes a 9-inch cake, what is the scale factor?

Given a point or figure and a scale factor, you will produce the dilated image, recover the scale factor from corresponding points, and decide whether size was preserved.

Definitions and rules

Dilation centred at the origin

D(k)(x, y) = (kx, ky)

k is the scale factor. k > 1 enlarges; 0 < k < 1 shrinks.

Why it has to be this: multiplying both coordinates by the same number scales every distance from the origin equally.

Why this one is not a rigid motion

Take P(2, 1) and Q(6, 4). The distance PQ is 5. Apply D(2) and the distance becomes 10.

distance is multiplied by k, not preserved

Angles are preserved, so the image is the same shape. Same shape, different size — similar, not congruent. Every transformation before this one was congruent.

Worked examples

Example 1 — recovering a scale factor and completing a triangle

A dilation centered at the origin maps A(2, 1) to A′(3, 3/2). The same dilation acts on B(6, 1) and C(2, 5). Find the scale factor and the images of B and C.

Suggested work

Givens

A(2, 1) → A′(3, 3/2) B(6, 1) C(2, 5)

Tools

D(k)(x, y) = (kx, ky)

Primitives

2k = 3, so k = 3/2

Computation

B(6, 1) → ((3/2)(6), (3/2)(1)) = B′(9, 3/2) C(2, 5) → ((3/2)(2), (3/2)(5)) = C′(3, 15/2)

Notice: One corresponding nonzero coordinate fixes the scale factor; the same factor must work for every coordinate.

Why one factor controls the whole figure

A dilation centered at the origin multiplies every distance from the origin by the same factor. Changing the factor from one vertex to another would not be one dilation.

Example 2 — enlarging a figure

Apply D(3) to A(1, 2), B(4, 2), C(1, 5).

Suggested work

Givens

A(1, 2) B(4, 2) C(1, 5) k = 3, centred at the origin

Tools

D(k)(x, y) = (kx, ky)

Primitives

k = 3 ← fixed for the whole figure

Computation

A(1, 2) → (3(1), 3(2)) = A′(3, 6) B(4, 2) → (3(4), 3(2)) = B′(12, 6) C(1, 5) → (3(1), 3(5)) = C′(3, 15)

Notice: every coordinate tripled. The shape is unchanged; the size is not.

Example 3 — one point enlarged by a factor of 2

Apply D(2) to P(2, 3).

Suggested work

Givens

P(2, 3) k = 2

Tools

D(k)(x, y) = (kx, ky)

Primitives

x = 2 y = 3 k = 2

Computation

D(2)(2, 3) = (2(2), 2(3)) = P′(4, 6)

Why both coordinates use 2

Multiplying both coordinates by the same factor scales the point's distance from the origin without changing its direction from the origin.

Common traps

Congruent or similar

Translations, rotations and reflections all give a congruent image — same size, same shape. A dilation gives a similar image — same shape, different size. If a question asks whether two figures are congruent and a dilation was involved, the answer is no unless the scale factor is 1.

Practice

Apply D(4) to P(2, 3).
Answer — tap to show
P′(8, 12)
Suggested work — tap to show
Givens
P(2, 3) k = 4
Tools
D(k)(x, y) = (kx, ky)
Primitives
x = 2 y = 3 k = 4
Computation
= (4(2), 4(3)) = (8, 12)
Apply D(3) to R(1, 2).
Answer — tap to show
R′(3, 6).
Suggested work — tap to show
Givens
R(1, 2) k = 3
Tools
D(k)(x, y) = (kx, ky)
Primitives
x = 1 y = 2 k = 3
Computation
= (3(1), 3(2)) = (3, 6)
Apply D(½) to Q(6, −8).
Answer — tap to show
Q′(3, −4)
Suggested work — tap to show
Givens
Q(6, −8) k = ½
Tools
D(k)(x, y) = (kx, ky)
Primitives
x = 6 y = −8 k = ½
Computation
= (½(6), ½(−8)) = (3, −4) 0 < k < 1, so the figure shrank.
Apply D(1/3) to Q(−9, 6).
Answer — tap to show
Q′(−3, 2).
Suggested work — tap to show
Givens
Q(−9, 6) k = 1/3
Tools
D(k)(x, y) = (kx, ky)
Primitives
x = −9 y = 6 k = 1/3
Computation
= ((1/3)(−9), (1/3)(6)) = (−3, 2)
A dilation centered at the origin maps A(4, −6) to A′(10, −15). Find the scale factor and decide whether the dilation preserves congruence for a figure with a nonzero side length.
Answer — tap to show
The scale factor is 5/2. It does not preserve congruence because every nonzero length is multiplied by 5/2.
Suggested work — tap to show
Givens
A(4, −6) → A′(10, −15)
Tools
D(k)(x, y) = (kx, ky)
Primitives
4k = 10
Computation
k = 10(1/4) = 5/2 Check: (−6)(5/2) = −15 A nonzero length L maps to (5/2)L, which is not L.