Enlargement
For positive scale factors, an enlargement maps each point along the ray from the centre through that point. The factor compares image and original lengths; angles remain unchanged
- Form the scale factor Divide an image length by the corresponding original length \(\displaystyle k=\frac{\text{image length}}{\text{original length}}\)
- Read the value of k A value greater than 1 enlarges a figure, while a value between 0 and 1 reduces it \(\displaystyle k=\frac{21}{14}=\frac{3}{2}>1\)
- Measure from the centre Each image point lies on the ray from the centre through the original point, at k times the original distance \(\displaystyle OA'=k\times OA\)
- Use the direction Multiply an original length by k to find its image length, or divide an image length by k to work backwards \(\displaystyle x=10\times\frac{3}{2}=15\)
- Identity At a scale factor of one, every point is unchanged. The centre cannot be uniquely recovered from two identical figures \(\displaystyle k=1\)
- Angles Enlargement and reduction preserve corresponding angles, even though side lengths change
Worked example
A segment of length 14 cm is enlarged to 21 cm. A second original segment is 10 cm long. Find the length of its image
- \(\displaystyle k=\frac{21}{14}=\frac{3}{2}\)The image length divided by the original length gives the scale factor
- \(\displaystyle x=10\times\frac{3}{2}=\frac{30}{2}\)Multiply the second original length by the scale factor
- \(\displaystyle x=15\ \mathrm{cm}\)The image of the second segment is 15 cm long
Common mistakes
Difficulty levels
Questions about this topic
How can the scale factor show whether a figure is enlarged or reduced?+
If k is greater than 1, the image is larger. If 0 is less than k and k is less than 1, the image is smaller
What is the role of the centre of enlargement?+
The centre fixes the direction of each image point. The original point, image point and centre lie on one straight line
How is an original length found from an image length?+
Divide the image length by the scale factor. This reverses the multiplication used to produce the image
Are correct distance ratios alone enough to find a centre?+
No. Corresponding points and the centre must also be collinear, with each image on the same ray for a positive factor. Use two different joining lines and check the factor
What happens when k equals 1?+
Every point is unchanged. The image and original coincide, so any centre gives the same identity transformation
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 7: Geometry and similarity