Similarity and Enlargement
Geometry · Similarity and Enlargement
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.3 Similarity and Enlargement (8 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Identify similar figures and state the conditions for similarity.
- Determine the linear scale factor between similar figures and use it to find unknown sides.
- Construct enlargements given a centre of enlargement and a scale factor.
- Relate the linear scale factor to the ratios of area and volume.
Similarity and Enlargement
A photograph printed small and printed large shows the same picture. Nothing is stretched or squashed. The two prints are similar.
This topic is about shapes that keep their proportions when their size changes.
a) Similar figures
Two figures are similar when they have the same shape but not necessarily the same size.
Similar figures obey two conditions at once. Matching angles are equal. Matching sides are all in the same ratio.
Both must hold. A square and a rectangle have equal angles. But their sides are not in a constant ratio, so they are not similar.
b) The linear scale factor
The linear scale factor is the number that multiplies every length on the object to give the image.
Read the rule carefully: image divided by object, in that order. Getting it the wrong way round gives 0.5 instead of 2.
A scale factor bigger than 1 makes the shape larger. A scale factor between 0 and 1 makes it smaller, but the shapes are still similar.
c) Finding a missing side
This is the most common question in the topic. You are given one matching pair of sides and asked for another.
Find the scale factor from the pair you know. Then apply it to the side you want.
Check that matching sides really do match. The longest side of the object pairs with the longest side of the image. Position on the page does not decide it.
d) Enlargement
An enlargement produces a similar figure from a fixed point called the centre of enlargement.
Draw a straight line from the centre through any corner of the object. The matching corner of the image lies on that same line. That is what makes the construction work.
To draw one, follow three steps. Join the centre to each corner of the object. Extend each line so it becomes the scale factor times as long. Join the new points up.
Two properties are worth noting. Angles do not change during an enlargement. Lengths are all multiplied by the same scale factor.
e) A caution about area
If lengths double, area does not double. It becomes four times larger, because area involves two lengths multiplied together.
You are not asked to calculate with area scale factors in Grade 9. But knowing that lengths and areas scale differently will stop you guessing wrongly.
f) Where this is used
An architect's drawing is a similar figure of a real building. A tailor grading a pattern from small to large is using a scale factor. A phone screen showing a photograph enlarges it without distorting faces.
Words to know
- Similar figures -- figures with equal corresponding angles and sides in a constant ratio.
- Linear scale factor -- the ratio of an image length to the corresponding object length.
- Centre of enlargement -- the fixed point from which rays are drawn to construct an enlargement.
- Image -- the figure produced by transforming the original, which is called the object.
- Congruent figures -- similar figures with a scale factor of exactly 1, so identical in size as well as shape.
:::checkpoint Check yourself
- What two conditions must hold for figures to be similar?
- An object side of 6 cm becomes 15 cm in the image. What is the scale factor?
- Two similar triangles have matching sides 4 cm and 10 cm. If another side of the small triangle is 6 cm, find the matching side of the large one.
- What happens to the angles of a shape during an enlargement? :::
Bridge to practice
The exercises begin with recognising similarity and finding linear scale factors, move through enlargement construction, and finish with area and volume ratios worked in both directions. Whenever a question supplies a ratio, pause to identify whether it describes lengths, areas or volumes before you use it, since that single decision determines the whole answer.