Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Dilations on the Cartesian Plane

E1.4 A1About 50 minutes
Learning goals
  • I can dilate a shape on the Cartesian plane using any centre and any scale factor, including fractions.
  • I can predict how a dilation changes side lengths, perimeter and area.
  • I can find the scale factor and the centre of a dilation from a shape and its image.
  • I can combine a dilation with other transformations.

Words to know

dilation
A transformation that enlarges or reduces a shape from a fixed point by a scale factor. The image is similar to the original.
centre of dilation
The fixed point the shape is enlarged or reduced from. It does not move.
scale factor (k)
The number every distance from the centre is multiplied by. k > 1 enlarges, 0 < k < 1 reduces, and k = 1 leaves the shape the same size.
similar
Same shape but not necessarily the same size: equal angles, and all side lengths multiplied by the same number.

Remember from Grade 7

In Grade 7 you dilated shapes using a centre and a scale factor, and when the centre was the origin you used the rule (x, y) → (kx, ky). You learned that the image is similar to the original: angles stay the same and every side is multiplied by k. In Grade 8 you'll dilate from any centre, see what happens to perimeter and area, and work backwards to find the centre and scale factor.

Let's start with a projector

When a teacher projects a picture onto a screen, light spreads out from the projector lens. Every point of the picture moves away from the lens along a straight line, and every distance is multiplied by the same amount. The lens is the centre of dilation. Move the projector back and the scale factor gets bigger.

xy−112345678−1123456CPQRP′Q′R′centre C(1, 1)scale factor 2PQ = 2 unitsP′Q′ = 4 units
Triangle PQR dilated by a scale factor of 2 from centre C(1, 1). Each image point is on the dashed line from C through the original point, twice as far from C. P(2, 2) → P′(3, 3), Q(4, 2) → Q′(7, 3), R(2, 3) → R′(3, 5). Side PQ is 2 units and P′Q′ is 4 units.

Dilating from any centre

When the centre C(a, b) is not the origin, use move, multiply, move back:

  1. Move: subtract the centre's coordinates from the point.
  2. Multiply: multiply both coordinates by the scale factor k.
  3. Move back: add the centre's coordinates again.

As a rule: (x, y) → (a + k(x − a), b + k(y − b)).

What a dilation does to a shape

  • Angles stay the same.
  • Every length (each side, and the perimeter) is multiplied by k.
  • Area is multiplied by k × k = k², because both the length and the width are multiplied by k.
  • Each side of the image is parallel to the matching side of the original.
  • The centre, a point and its image are always on one straight line.
Worked example

Example 1: A centre that is not the origin

Dilate triangle P(2, 2), Q(4, 2), R(2, 3) by a scale factor of 2 from the centre C(1, 1).

  1. Move (subtract C): P → (1, 1), Q → (3, 1), R → (1, 2).
  2. Multiply by 2: (2, 2), (6, 2), (2, 4).
  3. Move back (add C): P′(3, 3), Q′(7, 3), R′(3, 5).
  4. Check: PQ = 4 − 2 = 2 units and P′Q′ = 7 − 3 = 4 units. 4 ÷ 2 = 2 ✓

P′(3, 3), Q′(7, 3), R′(3, 5) (see the grid).

Worked example

Example 2: A reduction from the origin

Dilate triangle (−6, 4), (2, 8), (4, −2) by a scale factor of 1/2 from the origin.

  1. With the origin as centre, multiply each coordinate by 1/2 (the same as dividing by 2).
  2. (−6, 4) → (−3, 2). (2, 8) → (1, 4). (4, −2) → (2, −1).

(−3, 2), (1, 4), (2, −1). Each side of the image is half as long, so this is a reduction.

Worked example

Example 3: Perimeter and area

A rectangle is 3 cm by 2 cm. It is dilated by a scale factor of 3. Compare the perimeters and areas.

  1. Original: perimeter 2 × (3 + 2) = 10 cm; area 3 × 2 = 6 cm².
  2. Image: 9 cm by 6 cm. Perimeter 2 × (9 + 6) = 30 cm; area 9 × 6 = 54 cm².
  3. Perimeter: 30 ÷ 10 = 3, which is k. Area: 54 ÷ 6 = 9, which is k² = 3 × 3.

The perimeter is 3 times as big and the area is 9 times as big.

3 cm2 cmarea 6 cm²× 39 cm6 cmarea 54 cm² = 9 × 6 cm²k = 3lengths × 3area × 3² = × 9
Example 3 as a picture. With a scale factor of 3, the 3 cm by 2 cm rectangle becomes 9 cm by 6 cm. Nine copies of the original fit inside the image, because the area is multiplied by 3 × 3 = 9.
Worked example

Example 4: Find the scale factor and the centre

A dilation moves A(1, 1) to A′(3, 3) and B(2, 1) to B′(5, 3). Find the scale factor and the centre.

  1. Scale factor: AB = 2 − 1 = 1 unit and A′B′ = 5 − 3 = 2 units, so k = 2 ÷ 1 = 2.
  2. The centre is on the line through A and A′, and on the line through B and B′. Going from A′ to A is 2 left and 2 down, so the first line keeps going down-left: (1, 1), (−1, −1), ...
  3. Going from B′ to B is 3 left and 2 down, so continue from B: (2 − 3, 1 − 2) = (−1, −1).
  4. Both lines pass through (−1, −1). Check with move, multiply, move back: A(1, 1) → (2, 2) → (4, 4) → (3, 3) ✓

Scale factor 2, centre (−1, −1).

Worked example

Example 5: A dilation and a translation

Start with (3, 2). Plan A: dilate by 2 from the origin, then translate (x, y) → (x + 1, y − 3). Plan B: translate first, then dilate.

  1. Plan A: (3, 2) → (6, 4) → (7, 1).
  2. Plan B: (3, 2) → (4, −1) → (8, −2).
  3. In Plan B the translation is doubled too, because it happens before the dilation.

Plan A gives (7, 1) and Plan B gives (8, −2). The order matters.

Predict before you plot

  • k > 1: the image is bigger and farther from the centre.
  • 0 < k < 1: the image is smaller and closer to the centre.
  • A point at the centre doesn't move.
  • Area grows by k², not k. Doubling a photo's width and height makes it 4 times the area.
  • Check by measuring: image length ÷ original length should equal k for every side.
Hands-on activity

Flashlight dilations

You need: a flashlight, a wall, a small cardboard shape (like a triangle cut from a cereal box, with a grown-up's help with scissors), a ruler and a partner.

  1. In a dim room, hold the flashlight still. Hold the cardboard shape halfway between the flashlight and the wall, parallel to the wall.
  2. Your partner measures one side of the shape and the same side of the shadow. Find the scale factor: shadow length ÷ shape length.
  3. Move the shape closer to the flashlight. What happens to the scale factor? Move it closer to the wall.
  4. Measure the distance from the flashlight bulb to the shape and to the wall. Compare their ratio with the scale factor.

What to look for: the flashlight bulb acts like the centre of dilation. The shadow's scale factor is about (distance from bulb to wall) ÷ (distance from bulb to shape). Halfway gives a scale factor of about 2. (A flashlight's bulb isn't a perfect point, so expect the edges to be a little fuzzy and the numbers to be close, not exact.)

Math talk: "Where would you hold the shape to make the scale factor 3?"

Photocopier hack

Want a poster with twice the area of a page? Don't press 200% on the photocopier! That doubles the length and the width, so the area is 4 times as big and spills off the paper. Press about 141% instead, because 1.41 × 1.41 ≈ 2.

Your turn

1

Dilate (3, −2) by a scale factor of 4 from the origin.

2

Dilate (8, −6) by a scale factor of 1/2 from the origin.

3

Dilate (5, 3) by a scale factor of 2 from the centre (1, 1).

4

A side is 6 cm long. The shape is dilated by a scale factor of 1/3. How long is the image side?

5

A square has an area of 10 cm². It is dilated by a scale factor of 3. What is the area of the image?

6

Is a dilation with a scale factor of 0.75 an enlargement or a reduction?

7

A(2, 3) → A′(6, 9) in a dilation from the origin. What is the scale factor?

More practice: warm-up

1

Does a scale factor of 2 make the image larger or smaller?

2

Dilate (1, 4) by a scale factor of 3 from the origin.

3

Do the angles of a shape change in a dilation?

4

A side is 5 cm long. The scale factor is 2.5. How long is the image side?

5

What happens with a scale factor of 1?

More practice: on your own

1

Dilate the triangle (2, 2), (6, 2), (2, 4) by a scale factor of 1/2 from the centre (2, 2).

2

A rectangle is 4 cm by 6 cm. It is dilated by a scale factor of 1.5. Find the perimeter and area of the original and the image.

3

A dilation moves A(4, 2) to A′(7, 2) and B(4, 4) to B′(7, 5). Find the scale factor and the centre.

4

Dilate (−3, 6) by a scale factor of 2/3 from the origin.

5

Dilate (2, −1) by a scale factor of 3 from the origin, then reflect the image in the y-axis.

Challenge

1

A shape has an area of 3 cm². Its image after a dilation has an area of 75 cm². What is the scale factor?

2

A dilation with centre (2, 1) and scale factor 3 moves point P to (11, 7). Find P.

3

A shape is dilated by 2 from the origin, and the image is dilated by 3 from the origin. What single dilation does the same thing? What if the second scale factor were 1/2 instead?

Check your understanding

Quiz

1.

Dilate (4, −6) by a scale factor of 1/2 from the origin.

2.

A triangle has a perimeter of 12 cm. It is dilated by a scale factor of 3. What is the image's perimeter?

3.

A shape has an area of 5 cm². It is dilated by a scale factor of 2. What is the image's area?

4.

The image of a dilation is always:

5.

Dilate (3, 3) by a scale factor of 2 from the centre (1, 1).

6.

Which scale factor gives a reduction?

For parents and teachers

Big idea: The Ontario curriculum (E1.4) includes dilations on the Cartesian plane, along with translations, reflections and rotations, and asks students to predict the results. Grade 7 used the origin as the centre; Grade 8 uses any centre and looks at how perimeter and area change.

Common difficulties:

  • Multiplying the coordinates when the centre is not the origin. Remind your child: move, multiply, move back.
  • Thinking area is multiplied by k. Draw a 1 by 1 square and its image with k = 2: the image holds 4 of the small squares.
  • Confusing enlargement and reduction. k bigger than 1 enlarges; k between 0 and 1 reduces.

How to help: Photo zooming, projectors and shadows are all dilations. The flashlight activity makes the centre of dilation visible. Ask, "If we double the size of this photo, how much more paper will it need?" (4 times as much.) This lesson also builds A1 skills: making predictions and checking them.