Summary, Formula Sheet & Quick Quiz പുനരവലോകനം
All five key results of the chapter, the formulas you will use in problems, the mistakes to avoid, flashcards for the Malayalam terms, and a scored quiz to find out what needs another look.
1. The five key ideas
Same angles ⇒ the sides, taken in order of size (or opposite equal angles), are in the same ratio: p/a = q/b = r/c.
Same angles ⇒ all sides change by the same scale factor k: a = kp, b = kq, c = kr.
All sides scaled by the same factor ⇒ the triangles have the same angles.
If the sides are scaled by k, the area is scaled by k².
Two sides scaled by the same factor + equal included angle ⇒ the third side is scaled by the same factor and the other two angles are equal.
If two triangles satisfy any one of the three conditions below, they satisfy the other two as well. Such triangles are called similar.
2. Three ways to show two triangles are similar
Same angles (AA)
Two pairs of equal angles are enough – the third pair is then automatic (angle sum 180°).
All sides in the same ratio (SSS)
a : ka, b : kb, c : kc – check that all three ratios are equal.
Two sides in ratio + included angle (SAS)
The equal angle must be between the two scaled sides.
Congruent (തുല്യം) = similar with scale factor 1. Every congruent pair is similar; a similar pair is congruent only when k = 1.
3. Formula sheet
| Situation | Formula / fact | Where |
|---|---|---|
| Scale factor from triangle 1 to triangle 2 | k = (side of 2) ÷ (matching side of 1). Reverse direction: 1/k. | p. 4 |
| Matching sides | a = kp, b = kq, c = kr; pair sides opposite equal angles, or by size order. | p. 5 |
| Area of a scaled triangle | area × k² (sides ½ ⇒ area ¼; sides ×3 ⇒ area ×9). | p. 7 |
| Perpendicular from the right angle to the hypotenuse, cutting it into a and b | h² = ab; also (leg)² = (adjacent part) × (hypotenuse). | p. 6, Q2 & Q5 |
| Ropes between poles of heights a and b | crossing height h with 1/h = 1/a + 1/b, i.e. h = ab/(a + b); independent of the distance. | p. 6, Q6 |
| Square in a right triangle with legs a, b (vertex at the right angle) | side s = ab/(a + b) | p. 6, Q4 |
| Bisector of ∠A meets BC at P | BP/PC = AB/AC | p. 6, Q7 |
| Line parallel to a side of a triangle | cuts the other two sides in the same ratio; converse: same ratio ⇒ parallel. | p. 3 |
| Dividing a line AB in the ratio m : n | equal angles at A (above) and B (below); mark m and n units; join; the crossing point divides AB. | p. 6, Q3 |
| Enlarging any figure by k | from a point O send each vertex P to P′ with OP′ = k·OP (or use concentric circles for a triangle / rectangle / regular polygon). | p. 9 |
| Right triangles | two pairs of sides in the same ratio are enough (Pythagoras fixes the third). | p. 10 |
| Heights, medians, bisectors of similar triangles | scale by the same k as the sides. | p. 10 |
4. Common mistakes
- Pairing the wrong sides. Always pair sides opposite equal angles (or shortest ↔ shortest, longest ↔ longest). "AB with PQ" is not automatic – check the angle letters.
- Forgetting to square for areas. Sides × 3 means area × 9, not × 3.
- Mixing up k and 1/k. From 4 cm to 6 cm the factor is 6 ÷ 4 = 1½; from 6 back to 4 it is ⅔. Ask: "new ÷ old".
- Treating quadrilaterals like triangles. A square and a rectangle have the same angles but are not similar; sides 6, 9, 6, 9 can make a rectangle or a leaning parallelogram. Only triangles are fixed by their angles (or by their sides).
- SAS needs the included angle. Two sides in ratio plus an angle that is not between them proves nothing.
- Units. Convert everything to the same unit before taking a ratio (2 m and 50 cm ⇒ 200 : 50).
5. Flashcards പദങ്ങൾ
Click a card to flip it.
6. Scored quiz (10 questions)
Answer every question, then press Submit. Each wrong answer tells you which page to revisit.
7. Your study checklist
Pages you have marked as completed are ticked. Aim to tick them all before the model test.
If you scored 8 or more in the quiz and the checklist is complete, go to the Model Test Paper. Attempt it in 1½ hours without looking at the solutions.
Next: Model Test Paper ▶