Similar figures
Similar figures have the same shape. Their size may be different. Congruent figures have the same shape and the same size. So every congruent pair is also similar, but not every similar pair is congruent.
All circles are similar. All squares are similar. All equilateral triangles are similar. But two rectangles need not be similar: a long thin one and a fat one have different shapes.
When are two polygons similar?
Two polygons with the same number of sides are similar when (1) their matching angles are equal and (2) their matching sides are in the same ratio. Both conditions are needed. A square and a rhombus of the same side have equal side ratios but different angles, so they are not similar.
Similar triangles
ΔABC is similar to ΔPQR (written ΔABC ~ ΔPQR) when ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and
AB/PQ = BC/QR = CA/RP
The order of letters matters. A matches P, B matches Q, C matches R. Always write the matching letters in the same order.
The common ratio is called the scale factor. If it is 2, every side of the big triangle is 2 times the matching side of the small one.
Basic Proportionality Theorem (BPT)
Statement: If a line is drawn parallel to one side of a triangle and it cuts the other two sides at two different points, then it divides those two sides in the same ratio.
In ΔABC, if DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC. (This result is also called Thales' theorem.)
Proof idea (in simple words)
- Join BE and CD. Draw heights DM ⟂ AC and EN ⟂ AB.
- Area of ΔADE = ½ × AD × EN and area of ΔBDE = ½ × DB × EN. So ar(ADE)/ar(BDE) = AD/DB.
- In the same way, ar(ADE)/ar(DEC) = AE/EC.
- ΔBDE and ΔDEC stand on the same base DE and lie between the same parallels DE and BC, so their areas are equal.
- So AD/DB = AE/EC.
Other useful forms: AD/AB = AE/AC and DB/AB = EC/AC.
Converse of BPT
If a line cuts two sides of a triangle in the same ratio, then it is parallel to the third side. In the 3D, when AE : EC becomes equal to AD : DB, the line DE turns green (parallel).
Criteria for similarity: AA, SSS, SAS
You do not need to check all 6 facts (3 angles and 3 ratios). One of these tests is enough:
- AAA / AA: If two angles of one triangle are equal to two angles of another, the triangles are similar. (The third angles then match automatically, because angles add to 180°.)
- SSS: If all three pairs of matching sides are in the same ratio, the triangles are similar.
- SAS: If one angle of a triangle equals one angle of the other, and the sides that hold this angle are in the same ratio, the triangles are similar.
Careful: for SAS the angle must be the one between the two sides.
Try it yourself (practical)
Shadow test: On a sunny day, stand a 30 cm ruler straight on the ground and measure its shadow. At the same time measure the shadow of a pole or tree. Ruler/its shadow = pole/its shadow, because the sun's rays make similar triangles. Find the pole's height.
Paper test for BPT: Draw any triangle. Draw a line parallel to one side using a set square. Measure the four parts on the other two sides and divide. The two ratios come out almost equal.
In the 3D: on the last step, predict AD : DB for position 3, then slide and check.
Key formulas and definitions
- ΔABC ~ ΔPQR ⇔ ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and AB/PQ = BC/QR = CA/RP
- BPT: DE ∥ BC ⇒ AD/DB = AE/EC (also AD/AB = AE/AC)
- Converse of BPT: AD/DB = AE/EC ⇒ DE ∥ BC
- AA: two pairs of equal angles ⇒ similar
- SSS: AB/PQ = BC/QR = CA/RP ⇒ similar
- SAS: ∠A = ∠P and AB/PQ = AC/PR ⇒ similar
Worked examples
1. ΔABC ~ ΔPQR with AB = 4, BC = 6, CA = 5 and PQ = 8. Find QR and RP.
Scale factor = PQ/AB = 8/4 = 2. QR = 2 × 6 = 12. RP = 2 × 5 = 10.
2. In ΔABC, DE ∥ BC, AD = 2 cm, DB = 3 cm, AE = 4 cm. Find EC.
By BPT, AD/DB = AE/EC. 2/3 = 4/EC. EC = 4 × 3 ÷ 2 = 6 cm.
3. In ΔABC, D and E lie on AB and AC with AD = 3, DB = 6, AE = 2, EC = 4. Is DE ∥ BC?
AD/DB = 3/6 = 1/2. AE/EC = 2/4 = 1/2. The ratios are equal, so by the converse of BPT, DE ∥ BC.
4. Two triangles have angles 50°, 60°, 70° and 60°, 70°, 50°. Are they similar?
Yes. The angles match (50° with 50°, 60° with 60°, 70° with 70°), so by AA they are similar.
5. Sides of ΔABC are 3, 4, 6 and sides of ΔPQR are 9, 12, 18. Are they similar?
9/3 = 3, 12/4 = 3, 18/6 = 3. All ratios are equal, so the triangles are similar by SSS.
6. A 1.5 m tall girl casts a 2 m shadow. At the same time a pole casts a 12 m shadow. Find the height of the pole.
The sun's rays make the same angle, and both stand upright, so the triangles are similar (AA). Height/12 = 1.5/2. Height = 12 × 0.75 = 9 m.
7. In ΔABC, DE ∥ BC with AD = x, DB = x − 2, AE = x + 2, EC = x − 1. Find x.
By BPT, x/(x − 2) = (x + 2)/(x − 1). Cross-multiply: x(x − 1) = (x + 2)(x − 2). x² − x = x² − 4. So x = 4.
8. In ΔPQR, ∠P = 70°. In ΔXYZ, ∠X = 70°, PQ = 3, PR = 5, XY = 6, XZ = 10. Are they similar?
XY/PQ = 6/3 = 2 and XZ/PR = 10/5 = 2. The included angles ∠P = ∠X = 70°. So ΔPQR ~ ΔXYZ by SAS.
Common mistakes
- Mixing up the order of letters. In ΔABC ~ ΔPQR, AB matches PQ, not QR.
- Thinking similar means equal. Similar triangles can be of different sizes; congruent ones are equal.
- Using SAS with an angle that is not between the two sides.
- In BPT writing AD/AB = AE/EC. Keep the same kind of parts on both sides: AD/DB = AE/EC or AD/AB = AE/AC.