Similar shapes are enlargements of each other: every angle matches and every length is multiplied by the same scale factor. Find that factor once and every missing length follows.
In similar shapes, corresponding angles are equal and corresponding lengths are in the same ratio. Scale factor \(k = \dfrac{\text{length on new shape}}{\text{matching length on old shape}}\); multiply any old length by \(k\) to get the new one.
Fig. 1: DE is parallel to BC, so triangle ADE is similar to triangle ABC. AB = 4 + 6 = 10 cm, so the scale factor is 10 ÷ 4 = 2.5 and x = 6 × 2.5 = 15 (drawn to scale)
Section 01
What makes shapes similar.
Use the lab to enlarge a shape by different scale factors. Watch the angles stay fixed while every side is multiplied by the same number. Then stretch the shape in one direction only and notice it is no longer similar.
Similar: corresponding angles equal AND corresponding sides in the same ratio. One shape is an enlargement of the other.
Congruent: similar with scale factor 1, so identical in size too.
Scale factor: \(k = \text{new length} \div \text{old length}\), using a matching pair of sides. \(k > 1\) enlarges, \(0 < k < 1\) shrinks.
Always similar: any two circles, any two squares, any two equilateral triangles. Two rectangles are similar only if their sides are in the same ratio.
Worked example Testing for similarity
Rectangle P is 4 cm by 6 cm. Rectangle Q is 10 cm by 15 cm. Rectangle R is 6 cm by 8 cm. Which of Q and R is similar to P?
P to Q: \(10 \div 4 = 2.5\) and \(15 \div 6 = 2.5\). Same factor, so Q is similar to P.
P to R: \(6 \div 4 = 1.5\) but \(8 \div 6 = 1.33...\). Different factors, so R is not similar to P.
All the angles are 90° in every rectangle, so for rectangles only the side ratios decide.
Answer: Q is similar to P (scale factor 2.5); R is not.
Worked example A missing length
Two similar quadrilaterals: a side of 5 cm on the smaller one matches a side of 8 cm on the larger. Another side of the smaller one is 7.5 cm. Find the matching side on the larger one.
Scale factor \(k = 8 \div 5 = 1.6\).
Larger side \(= 7.5 \times 1.6 = 12\) cm.
Answer: 12 cm
Examiner trap
Adding the same amount does not make shapes similar. A 4 by 6 rectangle with 3 cm added to each side becomes 7 by 9, and \(7 \div 4 \ne 9 \div 6\). Similarity is about MULTIPLYING.
Section 02
Similar triangles.
In the similar-triangles tab, slide the parallel line up and down inside the big triangle. The small triangle on top keeps the same angles, so it stays similar, and the lab shows the scale factor changing as you move.
Nested triangles (Fig. 1): a line parallel to one side cuts off a small triangle similar to the big one. Corresponding angles are equal because of the parallel lines, and the angle at A is shared.
Crossed triangles (an X shape): two lines cross between two parallel lines. Vertically opposite angles are equal and alternate angles are equal, so the two triangles are similar.
Showing triangles are similar (Extended): give two pairs of equal angles, each with a reason (common angle, corresponding angles, alternate angles, vertically opposite angles). The third pair then must match too.
Match the sides carefully: corresponding sides sit opposite equal angles. Redraw the two triangles separately, the same way round, before you calculate.
Worked example Nested triangles
In Fig. 1, DE is parallel to BC, AD = 4 cm, DB = 6 cm and DE = 6 cm. Find BC.
Triangles ADE and ABC are similar: angle A is common and angle ADE = angle ABC (corresponding angles).
Use whole sides from A: AB = AD + DB = 4 + 6 = 10 cm (not 6 cm).
Scale factor \(k = AB \div AD = 10 \div 4 = 2.5\).
BC \(= DE \times k = 6 \times 2.5 = 15\) cm.
Answer: BC = 15 cm
Worked example Crossed triangles (Extended reasons)
AB is parallel to DE. Lines AE and BD cross at C. AC = 6 cm, CE = 9 cm and AB = 8 cm. Show that triangles ABC and EDC are similar, and find DE.
Angle BAC = angle DEC (alternate angles, AB parallel to DE).
Two pairs of equal angles, so the triangles are similar, with A matching E, B matching D and C matching C.
Scale factor from ABC to EDC: \(k = CE \div CA = 9 \div 6 = 1.5\).
DE matches AB, so DE \(= 8 \times 1.5 = 12\) cm.
Answer: Similar (two pairs of equal angles); DE = 12 cm
Your turn
In the crossed triangles above, BC = 5 cm. Find CD.
CD matches CB, so CD \(= 5 \times 1.5 = 7.5\) cm.
Examiner trap
In nested triangles the scale factor uses the WHOLE side AB, not the extra piece DB. Using \(6 \div 4\) here gives 9 cm for BC, which is wrong.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Two shapes are similar. Which statement must be true?
Quick check+10 XP
A side of 4 cm is enlarged to 10 cm. What is the scale factor?
Quick check+10 XP
A rectangle 3 cm by 5 cm is enlarged so that the 3 cm side becomes 12 cm. How long is the other side?
Quick check+10 XP
Which pair of shapes is ALWAYS similar?
Section 03
Similarity in context.
Shadows, photos, models and ramps all hide similar triangles or similar shapes. Spot the matching lengths, find the scale factor, then multiply.
Worked example Shadows
At the same time of day, a girl 1.8 m tall casts a shadow 2.4 m long and a tree casts a shadow 14 m long. Find the height of the tree.
The sun's rays make the same angle with the ground for both, so the two right-angled triangles are similar.
Scale factor from the girl to the tree: \(k = 14 \div 2.4 = 5.833...\).
Tree height \(= 1.8 \times 5.833... = 10.5\) m.
Answer: 10.5 m
Your turn
A photo 10 cm wide and 15 cm high is enlarged to be 24 cm wide. How high is the enlargement?
\(k = 24 \div 10 = 2.4\), so height \(= 15 \times 2.4 = 36\) cm.
Examiner trap
Keep each ratio the same way round. Height over shadow for the girl equals height over shadow for the tree: \(\dfrac{h}{14} = \dfrac{1.8}{2.4}\). Mixing a height with a shadow on one side gives nonsense.
Stage 2 · Exam
Exam-style questions
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Triangle PQR is similar to triangle XYZ, with P matching X, Q matching Y and R matching Z. PQ = 6 cm, QR = 9 cm and XY = 10 cm. Find YZ.
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In triangle ABC, D is on AB and E is on AC with DE parallel to BC. AD = 5 cm, DB = 3 cm and DE = 7.5 cm. Find BC.
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A post 2.5 m tall casts a shadow 1.5 m long. At the same time a building casts a shadow 21 m long. How tall is the building?
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AB is parallel to CD. Lines AC and BD cross at E. AE = 4 cm, EC = 10 cm and AB = 6 cm. Find CD.
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Similar shapes, scaled.
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