Similar triangles: criteria and properties
Similar triangles have the same shape: equal angles and proportional sides. Test with AA, SAS or SSS. Area ratio = (side ratio)².
The three similarity criteria
| Criterion | What you must show | Example |
|---|---|---|
| AA | Two angles of one triangle equal two angles of the other. The third angles are then equal too. | ∠A = ∠P and ∠B = ∠Q, so ΔABC ~ ΔPQR |
| SAS | One pair of angles equal, and the sides that include these angles in the same ratio. | ∠A = ∠P and AB/PQ = AC/PR |
| SSS | All three pairs of corresponding sides in the same ratio. | Sides 3, 4, 5 cm and 6, 8, 10 cm: every ratio is 1/2 |
Congruent vs similar
All congruent triangles are similar, but similar triangles need not be congruent. Write the letters in matching order: ΔABC ~ ΔPQR means A ↔ P, B ↔ Q and C ↔ R.
Properties of similar triangles
- If ΔABC ~ ΔPQR, then AB/PQ = BC/QR = CA/RP. This common ratio is the scale factor.
- The perimeters, altitudes (heights) and medians are in the same ratio as the sides.
- The areas are in the ratio of the squares of corresponding sides: ar(ΔABC)/ar(ΔPQR) = AB²/PQ².
- In a right triangle, the perpendicular from the right angle to the hypotenuse makes two triangles. They are similar to each other and to the whole triangle.
- Two triangles with a common vertex and bases on the same line have areas in the ratio of their bases.
Example 1: height from a shadow (word problem)
At 4 p.m., a boy 1.6 m tall casts a shadow 2.4 m long. At the same time, the flagpole (kodimaram) of a temple casts a shadow 12 m long. Find the height of the flagpole.
- The sun's rays fall at the same angle on both. Both stand at 90° to the ground.
- So the two triangles (height, shadow, ray) are similar by AA.
- Corresponding sides are proportional: h/1.6 = 12/2.4
- h = 1.6 × 12/2.4
- h = 1.6 × 5 = 8
The flagpole is 8 m tall.
Example 2: ratio of areas
ΔABC ~ ΔPQR. The area of ΔABC is 49 cm² and the area of ΔPQR is 81 cm². If QR = 13.5 cm, find BC.
- Area theorem: ar(ΔABC)/ar(ΔPQR) = (BC/QR)²
- (BC/QR)² = 49/81
- BC/QR = 7/9 (take the positive square root)
- BC = (7/9) × 13.5
- BC = 7 × 1.5 = 10.5
BC = 10.5 cm
A house plan is similar to the real house, drawn to a scale. You can find the height of a temple gopuram or a coconut tree from its shadow, without climbing it.
1 marks: “ΔABC ~ ΔDEF with AB = 3 cm and DE = 6 cm. Find ar(ΔABC) : ar(ΔDEF).” Side ratio is 1 : 2, so the area ratio is 1² : 2² = 1 : 4.
2 marks: “Are the triangles with sides 4 cm, 6 cm, 8 cm and 6 cm, 9 cm, 12 cm similar?” 4/6 = 6/9 = 8/12 = 2/3. All three ratios are equal, so the triangles are similar by SSS.
5 marks: “A boy 1.5 m tall casts a 2 m shadow. At the same time a tower casts a 30 m shadow. Find the height of the tower.” Same sun angle and both vertical, so the triangles are similar by AA. h/1.5 = 30/2, so h = 1.5 × 15 = 22.5 m.
AA, SAS, SSS: for similarity, match angles or ratios, not just lengths. Areas go by the square of the side ratio.