Last reviewed: 7 November 2026
Updated November 2026. These Grade 9 similarity and enlargement notes explain how to recognise similar figures, the properties of an enlargement, and how to find and use the linear scale factor. Two figures are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. An enlargement makes a similar copy of a shape from a fixed centre, and the linear scale factor tells you how many times longer each side of the image is than the matching side of the object.
The examples and questions are original practice for Kenyan junior school learners, not an official paper. For the other Grade 9 maths topics, see our Grade 9 maths revision guide and the Grade 9 notes hub.
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What the Grade 9 Curriculum Covers
Similarity and enlargement is a sub-strand under Geometry in the Kenya Institute of Curriculum Development’s Grade 9 Mathematics design. It covers similar figures and their properties, drawing similar figures, properties of enlargement, applying enlargement to draw objects and images, and the linear scale factor. The design is listed on the KICD Grade Nine curriculum designs page. Area and volume scale factors are not listed in this sub-strand, so these notes focus on lengths.
What Makes Two Figures Similar?
Similar figures have the same shape but not necessarily the same size. Both conditions must hold:
- Corresponding angles are equal.
- Corresponding sides are in the same ratio.
OpenStax Contemporary Mathematics’ section on triangles uses the same definition for similar triangles. For triangles, equal angles are enough to guarantee the sides are in proportion. For other shapes you need to check both. Every square is similar to every other square, but a square and a non-square rectangle are not similar even though all their angles are 90°, because their sides are not in the same ratio.
Worked example: are the rectangles similar?
Rectangle P is 4 cm by 6 cm. Rectangle Q is 6 cm by 9 cm. Ratios: 6 ÷ 4 = 1.5 and 9 ÷ 6 = 1.5. The ratios match and all angles are 90°, so P and Q are similar. Rectangle R is 5 cm by 6 cm: 5 ÷ 4 = 1.25 but 6 ÷ 6 = 1, so R is not similar to P.
Finding a Missing Side in Similar Figures
Triangle ABC is similar to triangle XYZ, with AB = 6 cm, BC = 8 cm and XY = 9 cm. Find YZ.
- Scale factor from ABC to XYZ: 9 ÷ 6 = 1.5.
- YZ = 8 × 1.5 = 12 cm.
Match the sides in the right order. The letters in “ABC is similar to XYZ” tell you that A matches X, B matches Y and C matches Z.
Properties of an Enlargement
An enlargement is described by a centre and a linear scale factor k.
- The object and its image are similar.
- Angles stay the same; lengths are multiplied by k.
- Each image point lies on the straight line from the centre through the object point.
- Distance from centre to image point = k × distance from centre to object point.
- If k is greater than 1 the image is bigger; if k is between 0 and 1 the image is smaller (this is still called an enlargement in maths).
Worked example: enlarging on a grid
Triangle with vertices A(1, 1), B(3, 1) and C(1, 2) is enlarged with centre the origin (0, 0) and scale factor 3. Multiply each coordinate by 3: A′(3, 3), B′(9, 3) and C′(3, 6). Side AB was 2 units; A′B′ is 6 units, three times longer. When the centre is the origin, multiplying the coordinates works. For any other centre, measure from the centre along the guide lines.
Worked example: finding the centre
If you are given an object and its image, join each object point to its image point with a straight line and extend the lines. The point where they meet is the centre of enlargement. Measure one pair of distances from the centre to get k.
The Linear Scale Factor
Linear scale factor = length on the image ÷ matching length on the object. A photo 10 cm wide printed 25 cm wide has k = 25 ÷ 10 = 2.5, so a 4 cm feature on the original becomes 4 × 2.5 = 10 cm. Map scales use the same idea: in Scale Drawing you met ratios such as 1 : 50,000, which is a scale factor between real and drawn lengths.
Original Practice Questions With Worked Answers
- A triangle has sides 3, 4 and 5 cm. A similar triangle has shortest side 9 cm. Find its other sides. k = 3, so 12 cm and 15 cm.
- Are a 2 cm by 5 cm rectangle and a 4 cm by 8 cm rectangle similar? 4 ÷ 2 = 2 but 8 ÷ 5 = 1.6. No.
- The point (2, −1) is enlarged with centre (0, 0) and scale factor 4. Find its image. (8, −4).
- A model car is 12 cm long. The real car is 4.8 m long. Find the scale factor from model to car. 4.8 m = 480 cm; 480 ÷ 12 = 40.
- An object is 5 cm from the centre of enlargement and k = 0.5. How far is the image point from the centre? 2.5 cm.
Cover the answers and attempt each question first. For timed practice, use the Grade 9 revision papers or the Grade 9 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
What are similar figures?
Figures with the same shape: corresponding angles are equal and corresponding sides are in the same ratio.
How do you find the linear scale factor?
Divide a length on the image by the matching length on the object.
What happens when the scale factor is less than 1?
The image is smaller than the object but still similar to it, with all lengths multiplied by the scale factor.
How do you find the centre of enlargement?
Draw straight lines through each object point and its image point. The point where the lines meet is the centre.