Proportionality & Similarity Theorems — Grade 12 (CAPS)

The proportion theorem for a line parallel to a side of a triangle, the two ways to prove triangles similar (equiangular, or sides in proportion), and how to lay out a proportion so the ratios cannot be flipped.

1The proportion theorem

A line drawn parallel to one side of a triangle divides the other two sides proportionally.

DE ∥ BC, so AD/DB = AE/EC.

The reason to quote in an exam is (line ∥ one side of △) — or 'prop theorem; DE ∥ BC'. Without the reason the statement earns nothing.

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Set every proportion out the same way round: top piece over bottom piece on both sides. Mixing AD/DB with EC/AE is the commonest error in the whole topic.

2Similar triangles

△ABC ||| △DEF means
Â = D̂, B̂ = Ê, Ĉ = F̂ and AB/DE = BC/EF = AC/DF

In Grade 12 there are two ways to prove that two triangles are similar:

Equiangular (AAA): the corresponding angles are equal. You only need to prove two pairs of angles equal — the third follows because the angles of a triangle add up to 180°. Reason: (∠∠∠) or (equiangular △s).
Sides in proportion: all three pairs of corresponding sides are in the same ratio. Reason: (sides of △ in prop).

Each one gives the other: equiangular triangles have their sides in proportion, and triangles with their sides in proportion are equiangular. Choose the route the question hands you — parallel lines and shared angles point to AAA; three given lengths point to the sides.

Worked ExampleIn △ABC, DE ∥ BC. Prove that △ADE ||| △ABC.

1
Â is common to both triangles.
Always look for the shared angle first.
2
AD̂E = AB̂C (corresponding ∠s; DE ∥ BC)
Parallel lines give equal corresponding angles.
3
AÊD = AĈB (corresponding ∠s; DE ∥ BC)
4
∴ △ADE ||| △ABC (AAA)
Write the letters in matching order — A with A, D with B, E with C.

Worked Example△PQR has PQ = 6, QR = 8 and PR = 10. △STU has ST = 9, TU = 12 and SU = 15. Prove that △PQR ||| △STU.

1
Pair the sides shortest with shortest: ST/PQ = 9/6 = 3/2.
Corresponding sides are matched by size, and in the same order as the letters.
2
TU/QR = 12/8 = 3/2 and SU/PR = 15/10 = 3/2.
All THREE ratios must be checked — two equal ratios are not enough.
3
∴ △PQR ||| △STU (sides of △ in prop)
The sides are in the same ratio, so the triangles are similar — and therefore equiangular too.

3Using similarity to find a length

Worked ExampleDE ∥ BC with AD = 4, DB = 6 and AE = 5. Calculate EC.

1
AD/DB = AE/EC.
Proportion theorem.
2
4/6 = 5/EC.
Substitute.
3
4 × EC = 30.
Cross-multiply.
4
EC = 7,5.
Check: 4/6 = 5/7,5 = 2/3. ✓

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AD = 4 and DB = 6 means AB = 10, not 6. Read carefully whether a question gives you the piece or the whole side — AD/AB = 4/10 is a different ratio from AD/DB = 4/6.

4Proving Pythagoras with similar triangles

The ATP asks you to prove the Theorem of Pythagoras using similarity. Drop a perpendicular from the right angle to the hypotenuse and three similar triangles appear at once.

△ABC with Â = 90° and AD ⊥ BC. This single line creates three similar triangles.

Worked ExampleProve that AB² + AC² = BC²

1
In △ABD and △CBA: B̂ is common, and AD̂B = BÂC = 90°.
Two pairs of equal angles.
2
∴ △ABD ||| △CBA (AAA), so AB/CB = BD/BA.
Corresponding sides in proportion.
3
∴ AB² = BD · BC.
Cross-multiply.
4
Similarly △ACD ||| △BCA gives AC² = CD · CB.
The same argument on the other side.
5
Add: AB² + AC² = BC(BD + CD).
Take BC out as a common factor.
6
But BD + CD = BC, so AB² + AC² = BC². □
D lies on BC, so the two pieces make the whole.

The same three similar triangles give a third result. From △ABD ||| △CAD: AD/CD = BD/AD, so AD² = BD · DC — the perpendicular from the right angle is the geometric mean of the two pieces of the hypotenuse.

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Check all three on a 3-4-5 triangle (AB = 3, AC = 4, BC = 5): BD = 1,8 and CD = 3,2. Then BD × BC = 1,8 × 5 = 9 = AB², CD × CB = 3,2 × 5 = 16 = AC², and BD × DC = 1,8 × 3,2 = 5,76 = 2,4² = AD². ✓

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
In △ABC below, DE ∥ BC. Write down the proportion that follows, with a reason.
Show answer ▾Hide answer ▴
AD/DB = AE/EC (line ∥ one side of △)
2
DE ∥ BC with AD = 4, DB = 6 and AE = 5. Calculate EC.
Show answer ▾Hide answer ▴
4/6 = 5/EC → EC = 7,5
3
In △ABC below with DE ∥ BC, prove that △ADE ||| △ABC.
Show answer ▾Hide answer ▴
Â common; AD̂E = AB̂C and AÊD = AĈB (corresponding ∠s, DE ∥ BC) → △ADE ||| △ABC (AAA)
4
If AD = 4 and DB = 6, write down the length of AB.
Show answer ▾Hide answer ▴
4 + 6 = 10
5
How many pairs of equal angles must you prove for AAA?
Show answer ▾Hide answer ▴
Two — the third follows because the angles of a triangle add to 180°.
6
△ABC ||| △DEF with AB = 8, DE = 12 and BC = 10. Calculate EF.
Show answer ▾Hide answer ▴
8/12 = 10/EF → EF = 15
7
State the reason you must quote when using the proportion theorem.
Show answer ▾Hide answer ▴
(line ∥ one side of △)
8
Two triangles are equiangular. What can you conclude?
Show answer ▾Hide answer ▴
They are similar, so their corresponding sides are in proportion.
9
The sides of △ABC are 4 cm, 6 cm and 8 cm. The sides of △KLM are 6 cm, 9 cm and 12 cm. Are the triangles similar? Give a reason.
Show answer ▾Hide answer ▴
6/4 = 9/6 = 12/8 = 1,5 ✓✓, so the sides are in proportion and the triangles are similar (sides of △ in prop) ✓