Year 10 Extended · Unit 5 Solids, similarity and transformations · Lesson 4 of 6

Enlarging from a point, on either side

A dilation moves every point along the line through a fixed centre, by a scale factor. This lesson asks what a fraction or a negative number does as a scale factor, and finds images, centres and scale factors.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: enlarge the point (3, 2) by scale factor 4 from the origin. Where is the image?
    Answer(12, 8). Each coordinate is multiplied by 4.
  2. From the last lesson: a shape of area 5 cm² is enlarged by scale factor 3. What is the new area?
    Answer45 cm², because 3² = 9 and 5 × 9 = 45.
  3. From Year 9: the point (2, −5) is reflected in the x-axis. Where is the image?
    Answer(2, 5). The y-coordinate changes sign.
Words used in this lesson
dilation
a transformation that multiplies every distance from a centre by the same scale factor. For example: a photo enlarged from its centre.
centre of dilation
the one point that does not move. For example: the pinhole of a camera.
scale factor
the number the distances from the centre are multiplied by. For example: 3 makes every distance from the centre three times as long.
image
the point or shape after the transformation. For example: P′ is the image of P.
opposite side
on the other side of the centre, on the same line. For example: (−1, 0) is on the opposite side of the origin from (3, 0).

The image that comes out upside down

In a pinhole camera built in a school in Nairobi, light from a candle passes through a tiny hole and falls on a screen. Each point of the flame sends a straight line of light through the hole, and the image of that point is where the line meets the screen. The hole acts as the centre of a dilation.

Predict first: is the image on the same side of the hole as the candle, and is it bigger or smaller? Use the figure to see what a scale factor does when it is smaller than 1, and what happens when it is negative.

0
0
1
(1, 1)image of P (1, 1)
(4, 1)image of Q (4, 1)
1area of the image ÷ area of the original

Centre (0, 0), scale factor 1: P (1, 1) goes to (1, 1) and Q (4, 1) to (4, 1). The area is multiplied by 1.

Show me the steps
  1. Keep the centre at (0, 0) and set the scale factor to 1.5: P (1, 1) goes to (1.5, 1.5) and Q (4, 1) to (6, 1.5). The area is multiplied by 2.25.
  2. Keep the centre at (0, 0) and set the scale factor to −2: P goes to (−2, −2) and Q to (−8, −2). The image is on the other side of the centre, and the area is multiplied by 4, not −4.
  3. Move the centre to (−2, 2) and set the scale factor to 0.5: P goes to (−0.5, 1.5) and Q to (1, 1.5). The image is smaller, on the same side, and the area is multiplied by 0.25.

The rule. A dilation with centre C and scale factor k sends P to P' with P' = C + k(P − C): the vector from C to P is multiplied by k. The image lies on the line CP.

Size and side. If |k| > 1 the image is larger, and if 0 < |k| < 1 it is smaller. If k is positive the image is on the same side of C as P, and if k is negative it is on the opposite side. A dilation with k = −1 is a half turn about C.

Lengths and area. Lengths are multiplied by |k| and areas by k², which is never negative. The image is similar to the original, and congruent only when |k| = 1.

Finding the centre. The lines through each point and its image all pass through the centre, so two of them meet there. The scale factor is the ratio of an image length to the original, with a minus sign if the image is on the opposite side.

A negative scale factor does not make lengths or areas negative. A scale factor of −3 makes every length 3 times as long and every area 9 times as big, and it puts the image on the other side of the centre. The centre is not always the origin, so multiply the distance from the centre, not the coordinates.

Worked examples

1. Find the image of P (2, 4) under a dilation with centre C (−1, 3) and scale factor 2.

  1. P − C = (2 − (−1), 4 − 3) = (3, 1).
  2. Multiply by 2: (6, 2). Add C: (−1 + 6, 3 + 2) = (5, 5).

2. Find the image of P (8, −1) under a dilation with centre (2, 2) and scale factor −⅓.

  1. P − C = (6, −3).
  2. Multiply by −⅓: (−2, 1). Add C: (2 − 2, 2 + 1) = (0, 3). The image is smaller and on the other side of C.

3. P (2, 5) goes to P' (4, 7) and Q (6, 1) goes to Q' (10, 1). Find the scale factor and the centre.

  1. P to Q is (4, −4) and P' to Q' is (6, −6), so the scale factor is 6 ÷ 4 = 1.5, positive because the direction is the same.
  2. Rearranging P′ = C + k(P − C) gives C = (kP − P′) ÷ (k − 1) = ((1.5 × 2 − 4) ÷ 0.5, (1.5 × 5 − 7) ÷ 0.5) = (−2, 1).
  3. Check with Q: C + 1.5(Q − C) = (−2 + 12, 1 + 0) = (10, 1), which matches Q′.

Your go. The first step is given. Find the image of P (4, 0) under a dilation with centre (2, −1) and scale factor −3.

  1. P − C = (4 − 2, 0 − (−1)) = (2, 1).
  2. Now multiply by −3 and add the centre.
Check the answer

−3 × (2, 1) = (−6, −3), and adding C (2, −1) gives (−4, −4). The image is three times as far from the centre, on the other side.

Check before you go on

1. A dilation has centre (0, 0) and scale factor −2. Where does the point (3, −1) go?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A dilation has centre (2, 1) and scale factor 3. Find the x-coordinate of the image of the point (4, 2).

3. A dilation has centre (1, −2) and scale factor −½. Find the y-coordinate of the image of the point (7, 4).

Level 2 · Building

4. A dilation about the origin sends (6, −4) to (−9, 6). Find the scale factor, as a decimal.

5. A dilation with scale factor 3 sends (2, 1) to (8, 7). Find the x-coordinate of its centre.

6. A triangle of area 18 cm² is dilated with scale factor −⅓. Find the area of the image, in square centimetres.

Level 3 · Stretch

7. A dilation with centre (1, 1) and scale factor −2 sends the point B to B′ = (−5, 7). Find the x-coordinate of B.

8. A dilation about the origin with scale factor 2 is followed by a dilation about the origin with scale factor −3. A single dilation about the origin does the same job. Find its scale factor.

Something to think about

Dilate a point by scale factor 2 about the origin, then dilate the image by scale factor ½ about the point (4, 0). Try it for (1, 1) and for (3, 5). What single transformation does the same job, and why is it not a dilation?

One way to think about it

(1, 1) goes to (2, 2), then to (4, 0) + ½((2, 2) − (4, 0)) = (3, 1). (3, 5) goes to (6, 10), then to (4, 0) + ½(2, 10) = (5, 5). Both have moved by (2, 0). In general 2P goes to (4, 0) + ½(2P − (4, 0)) = P + (2, 0), a translation by (2, 0). The scale factors multiply to 1, so sizes do not change, but no point stays where it is, and every dilation keeps its centre fixed.

Exit ticket

9. A dilation about a centre C sends a triangle of area 12 cm² to an image of area 3 cm², on the opposite side of C. What is the scale factor?