Year 9 · Unit 6 Transformations · Lesson 4 of 6

What does a scale factor do to every point?

An enlargement from the origin multiplies both coordinates by the scale factor. Lengths change, angles do not, so the image is similar to the pre-image but not congruent.

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 Lesson 3. Turn the point (3, 5) through 180° about the origin.
    Answer(−3, −5). A half turn changes both signs.
  2. From Year 8, Unit 1, ratio. Simplify the ratio 12 : 8.
    Answer3 : 2. Divide both by 4.
  3. From Year 8, Unit 3, integers. Work out (−3) × 4.
    Answer−12. A negative times a positive is negative.
Words used in this lesson
enlargement
a change of size from a centre, by a scale factor. For example: a photograph printed larger.
scale factor
the number every length is multiplied by. For example: 2 doubles; 0.5 halves.
reduction
an enlargement with a scale factor between 0 and 1: the image is smaller. For example: scale factor 0.5.
similar
the same shape but not necessarily the same size. For example: a photograph and its enlargement.
congruent
exactly the same size and shape. For example: two identical stamps.

Blowing up a photograph

A print shop in Nairobi enlarges a small photograph so that every length is 3 times as long. On a grid with the centre of the enlargement at the origin, one corner of the photograph sits at C (1, 2). Each corner of the print is 3 times as far from the origin as the matching corner of the photograph, along the same line.

Predict first: with scale factor 3, where does C (1, 2) land? Then use the slider to test your answer and see what a scale factor below 1 does.

2
(2, 2)image of A, A′
(4, 2)image of B, B′
(2, 4)image of C, C′

With scale factor 2 about the origin, A (1, 1) lands on A′ (2, 2). Every coordinate is multiplied by 2.

Show me the steps
  1. Scale factor 2: A (1, 1) goes to (2, 2), B (2, 1) to (4, 2) and C (1, 2) to (2, 4). Every coordinate is multiplied by 2.
  2. Scale factor 1: every image is the point itself, so A (1, 1), B (2, 1) and C (1, 2) stay put. The triangle does not change.
  3. Scale factor 3: A goes to (3, 3), B to (6, 3) and C to (3, 6). Each corner is three times as far from the origin.
  4. Scale factor 0.5: A goes to (0.5, 0.5), B to (1, 0.5) and C to (0.5, 1). A scale factor below 1 makes the image smaller, a reduction.

An enlargement from the origin with scale factor k multiplies both coordinates by k: (x, y) → (kx, ky). Every length is multiplied by k, every angle stays the same, and each image point lies on the line from the centre through its pre-image point.

If k is more than 1 the image is bigger; if k is between 0 and 1 it is smaller (a reduction); if k = 1 nothing changes.

The image is similar to the pre-image, not congruent, unless k = 1. This is why an enlargement is a similarity transformation, not a congruence transformation.

What changes and what does not

Compare the pre-image with the image: sides, angles, area.

2
2every length is multiplied by
4area of the image (pre-image 1)
similar, not congruentthe two triangles are

With k = 2 the base 2 becomes 4 and the height 1 becomes 2. The area 1 becomes 4, which is 2 × 2. The right angle is still 90°.

Show me the steps
  1. k = 2: the base 2 becomes 4 and the height 1 becomes 2. The area 1 becomes 4, which is 2 × 2. The triangles are similar, not congruent.
  2. k = 1: the image sits on the pre-image and has the same size. The triangles are congruent as well as similar.
  3. k = 0.5: the base is 1, the height 0.5 and the area 0.25, which is 0.5 × 0.5. The image is smaller, a reduction.
  4. k = 2.5: the base is 5, the height 2.5 and the area 6.25, which is 2.5 × 2.5. The right angle is still 90°.

The scale factor multiplies; it does not add. Adding 2 to each coordinate gives a translation, not an enlargement. For k = 2, the point (3, 1) goes to (6, 2) by multiplying, but adding 2 would give (5, 3). And because area is length times length, areas are multiplied by k × k, not by k.

Worked examples

Enlarge P (2, −3) by scale factor 4 from the origin. Multiply both coordinates: 2 × 4 = 8 and −3 × 4 = −12. P′ (8, −12).

Enlarge the triangle (1, 1), (3, 1), (1, 4) by scale factor 2. Double every coordinate: (2, 2), (6, 2), (2, 8). The base 2 is now 4 and the height 3 is now 6.

Reduce Q (6, −4) by scale factor one half. Half of 6 is 3 and half of −4 is −2: Q′ (3, −2). Because k is below 1, the image is closer to the origin.

Find the scale factor when A (3, 2) goes to A′ (9, 6).

  1. Compare the x-coordinates: 9 ÷ 3 = 3.
  2. Check with the y-coordinates: 6 ÷ 2 = 3.
  3. The scale factor is 3. The same factor works for both, which is what an enlargement needs.

A photo 6 cm by 4 cm is enlarged by scale factor 2.5. The sides are 6 × 2.5 = 15 cm and 4 × 2.5 = 10 cm. The area goes from 24 cm² to 150 cm², which is 24 × 6.25: the scale factor squared, not just 2.5.

Your go. The first step is given. Enlarge T (−4, 2) by scale factor 3 from the origin.

  1. x: 3 × (−4) = −12.
  2. Find the new y, then write T′.
Check the answer

y: 3 × 2 = 6, so T′ is (−12, 6).

Check before you go on

1. The point (2, 5) is enlarged from the origin with scale factor 3. Where does it land?

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. The point (4, −6) is enlarged by scale factor 3 from the origin. Its image is (x, y). What is y?

3. The point (10, −8) is reduced by scale factor 0.5 (one half) from the origin. What is the x-coordinate of the image?

Level 2 · Building

4. A rectangular flag on a Dhaka rooftop has a long side of 4 units. It is enlarged by scale factor 2.5. How long is the long side of the image?

5. An enlargement from the origin sends P (2, 3) to P′ (8, 12). What is the scale factor?

6. Which transformation does not give an image congruent to the pre-image?

Level 3 · Stretch

7. A triangle of area 12 cm² is enlarged by scale factor 3. What is the area of the image, in cm²?

8. A shape is enlarged by scale factor 2 from the origin, and the image is then enlarged by scale factor 3 from the origin. One single enlargement from the origin does the same job. What is its scale factor?

Something to think about

In the second figure the area of the image is k times k times the pre-image's area. A friend says that doubling the lengths should double the area. Use the figure to show what is wrong with that.

One way to think about it

Doubling both the base and the height makes the area 2 × 2 = 4 times as big, not 2 times. Area is length times length, so the scale factor is used twice. With k = 2 the area went from 1 to 4.

Exit ticket

9. A triangle is enlarged by scale factor 3 from the origin. What stays the same?