Grade by Grade
E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Dilations and Similar Shapes

E1.3About 50 minutes
Learning goals
  • I can perform a dilation using a centre of dilation and a scale factor.
  • I can tell whether a dilation is an enlargement or a reduction.
  • I can dilate a shape on a coordinate grid with the origin as the centre by multiplying the coordinates.
  • I can explain why the image of a dilation is similar to the original shape.

Words to know

dilation
A transformation that makes a shape bigger or smaller without changing its shape.
centre of dilation
The fixed point that the shape is enlarged or reduced from. Every point moves along a straight line away from or toward it.
scale factor
The number that every distance from the centre (and every side length) is multiplied by.
enlargement
A dilation with a scale factor greater than 1. The image is bigger.
reduction
A dilation with a scale factor between 0 and 1. The image is smaller.
similar
Same shape but not necessarily the same size: matching angles are equal and matching sides are in the same ratio.
corresponding
Matching. Corresponding sides or angles are in the same position in the two shapes.

Remember from earlier grades

In Grade 5 you learned that congruent shapes have the same size and the same shape. In Grade 6 you translated, reflected and rotated shapes, and the image was always congruent to the original. A dilation is new: it changes the size but keeps the shape.

Let's start with a photo

When you zoom in on a photo, everything gets bigger, but a round face doesn't turn into a long oval. All the lengths grow by the same amount, so the picture keeps its shape. That's a dilation. A shadow puppet does the same thing: the closer your hand is to the flashlight, the bigger its shadow on the wall.

How to dilate a shape

You need a centre of dilation and a scale factor.

  1. Draw a straight line from the centre through each vertex.
  2. Measure the distance from the centre to the vertex.
  3. Multiply that distance by the scale factor. Mark the image vertex that far from the centre, on the same line.
  4. Join the image vertices. Label them with primes: A′, B′, C′.
  • Scale factor greater than 1: an enlargement.
  • Scale factor between 0 and 1 (like 1/2): a reduction.
  • Scale factor exactly 1: the image is the same size, so it is congruent.

Shortcut on a grid: when the centre is the origin, just multiply both coordinates by the scale factor: (x, y) → (k × x, k × y).

OABCA′B′C′scale factor 2centre: the origin OA(1, 2) → A′(2, 4)B(3, 2) → B′(6, 4)C(1, 3) → C′(2, 6)
Triangle ABC dilated by a scale factor of 2 with the origin O as the centre. Each image vertex is twice as far from O as the original, along the same dashed line. Every side of A′B′C′ is twice as long as the matching side of ABC: AB = 2 units and A′B′ = 4 units.

Why the image is similar

In a dilation:

  • Corresponding angles are equal. A right angle stays a right angle.
  • Every side is multiplied by the same scale factor, so the ratio of image side to original side is the same for every pair of sides.
  • Corresponding sides are parallel.

Shapes like this are called similar. Congruent shapes are similar too (scale factor 1), but similar shapes don't have to be congruent.

Worked example

Example 1: A reduction on a grid

Dilate rectangle P(2, 2), Q(6, 2), R(6, 4), S(2, 4) with the origin as the centre and a scale factor of 1/2.

  1. Multiply each coordinate by 1/2 (the same as dividing by 2).
  2. P′(1, 1), Q′(3, 1), R′(3, 2), S′(1, 2).
  3. Check the sides: PQ = 4 and P′Q′ = 2; QR = 2 and Q′R′ = 1. Every side is half as long ✓

P′(1, 1), Q′(3, 1), R′(3, 2), S′(1, 2). The image is a reduction: a 2 by 1 rectangle, similar to the 4 by 2 rectangle.

xy12345671234OPQRSP′Q′R′S′
Scale factor 1/2, centre at the origin: every image vertex is halfway along the dashed line from O. The 4 by 2 rectangle shrinks to a similar 2 by 1 rectangle.
Worked example

Example 2: Find the scale factor

A triangle has sides 4 cm, 5 cm and 6 cm. After a dilation, the 5 cm side is 15 cm. Find the scale factor and the other two image sides.

  1. Scale factor = image length ÷ original length = 15 ÷ 5 = 3.
  2. 4 × 3 = 12 cm and 6 × 3 = 18 cm.

Scale factor 3; the image has sides 12 cm, 15 cm and 18 cm.

Worked example

Example 3: Similar or not?

Are these triangles similar? (a) sides 3, 4, 5 and sides 6, 8, 10. (b) sides 3, 4, 5 and sides 6, 8, 12.

  1. Divide each image side by the matching original side.
  2. (a) 6 ÷ 3 = 2, 8 ÷ 4 = 2, 10 ÷ 5 = 2. All the same.
  3. (b) 6 ÷ 3 = 2, 8 ÷ 4 = 2, but 12 ÷ 5 = 2.4. Not the same.

(a) is similar (scale factor 2). (b) is not similar.

Worked example

Example 4: A centre that isn't the origin

Dilate the point P(5, 3) with centre C(1, 1) and scale factor 2.

  1. From C to P: 5 − 1 = 4 units right and 3 − 1 = 2 units up.
  2. Multiply by 2: 8 units right and 4 units up.
  3. Start at C(1, 1) and move 8 right and 4 up: (1 + 8, 1 + 4).

P′(9, 5). (Multiplying the coordinates of P only works when the centre is the origin.)

Check your dilation

  • The image vertex must lie on the line from the centre through the original vertex.
  • Measure one side of the image and one side of the original. Their ratio should equal the scale factor.
  • Angles stay the same. If an angle changed, something went wrong.
Hands-on activity

Shadow dilations

You need: a flashlight, a piece of cardboard, a ruler, a wall, tape. Ask a grown-up before cutting the cardboard.

  1. Cut a simple shape, like a triangle or an L, out of cardboard. Measure its sides.
  2. In a dark room, hold the shape between the flashlight and the wall. The light bulb acts as the centre of dilation.
  3. Tape a sheet of paper to the wall and have a partner trace the shadow.
  4. Measure the sides of the shadow. Divide each by the matching side of the cardboard shape. Are the ratios about the same? That's your scale factor.
  5. Move the shape closer to the flashlight and trace again. What happens to the scale factor?

What to look for: ratios that are nearly equal (small measuring errors are normal), and a bigger scale factor when the shape is closer to the light. Keep the shape parallel to the wall, or the shadow gets stretched and isn't a true dilation.

Just for fun

Every time you zoom in on a photo of your pet, you are doing a dilation. Your cat just got a scale factor of 3, and it is still not impressed.

Your turn

1

Dilate (2, 3) with the origin as centre and scale factor 3.

2

Dilate (8, 4) with the origin as centre and scale factor 1/4.

3

Is a dilation with scale factor 0.5 an enlargement or a reduction?

4

A side is 6 cm long. The scale factor is 2.5. How long is the image side?

5

A triangle with angles 40°, 60° and 80° is dilated by a scale factor of 3. What are the angles of the image?

6

A side is 4 cm in the original and 2 cm in the image. What is the scale factor?

7

Are a 2 by 3 rectangle and a 4 by 9 rectangle similar?

More practice: warm-up

1

Dilate (1, 4) with the origin as centre, scale factor 2.

2

Dilate (6, 0) with the origin as centre, scale factor 1/3.

3

Is a scale factor of 4 an enlargement or a reduction?

4

What happens to a shape dilated with a scale factor of 1?

5

A square has sides of 5 cm. It is dilated by a scale factor of 2. How long is each side of the image?

More practice: on your own

1

Dilate triangle A(2, 0), B(4, 0), C(2, 6) with the origin as centre and scale factor 1/2.

2

A rectangle is 3 cm by 5 cm. After a dilation, its long side is 12.5 cm. Find the scale factor and the short side of the image.

3

A 3 cm by 5 cm rectangle is dilated with a scale factor of 2. Compare the perimeters.

4

Are triangles with sides 6, 8, 10 and 9, 12, 15 similar? Explain.

5

With the origin as centre, the point (4, 2) is dilated to (10, 5). What is the scale factor?

Challenge

1

Dilate P(4, 5) with centre C(2, 1) and scale factor 3.

2

A 2 cm by 3 cm rectangle is dilated with a scale factor of 3. How many times bigger is the area of the image? Explain.

3

A photo is 10 cm by 15 cm. Can it be enlarged, without stretching, to exactly fill a 20 cm by 25 cm frame? Explain.

Check your understanding

Quiz

1.

Dilate (3, 5) with the origin as centre and scale factor 2.

2.

A dilation with a scale factor of 1/3 is ...

3.

What happens to the angles of a shape in a dilation?

4.

What does similar mean?

5.

A side of 4 cm becomes 10 cm in a dilation. What is the scale factor?

6.

Which pair of rectangles is similar?

For parents and teachers

Big idea: E1.3 asks students to perform dilations and describe the similarity between the image and the original shape. Similar shapes have equal corresponding angles and proportional corresponding sides.

How to help at home: Use grid paper. Draw a simple shape and ask your child to dilate it by 2 from the origin, then by 1/2. Ask, "How do you know the shapes are similar?" The shadow activity makes the idea of a centre of dilation real.

Common difficulties:

  • Adding instead of multiplying (turning (3, 5) into (5, 7) for a scale factor of 2).
  • Multiplying coordinates when the centre is not the origin. Then you must measure from the centre.
  • Thinking the angles get bigger too.
  • Calling shapes similar when only one pair of sides has the right ratio. All pairs must match.