📑 Topics on this page (Jump to Topic):
📌 1. Triangles
- Angles sum to 180°; exterior angle = sum of the two interior opposite angles.
- Sum of any two sides > third side.
- Pythagoras: hyp² = base² + perp². Triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
| Centre | Meeting of | Property |
|---|---|---|
| Centroid | Medians | Divides median 2 : 1 |
| Incentre | Angle bisectors | ∠BIC = 90° + A/2 |
| Circumcentre | Perpendicular bisectors | ∠BOC = 2A |
| Orthocentre | Altitudes | ∠BHC = 180° − A |
📘 Formulas:
- Ratio of areas of similar triangles = ratio of squares of corresponding sides
- Equilateral triangle: height = (√3/2)a, inradius = a/(2√3), circumradius = a/√3
📌 2. Polygons & Circles
📘 Formulas:
- Sum of interior angles of an n-gon = (n − 2) × 180°
- Each exterior angle of a regular polygon = 360°/n
- Number of diagonals = n(n − 3)/2
- Angle at centre = twice the angle at circumference; angle in a semicircle = 90°
- Opposite angles of a cyclic quadrilateral sum to 180°
- A tangent is perpendicular to the radius; tangents from an external point are equal
- Chord at distance d from centre: (chord/2)² + d² = r²
⚠️ Exam Traps:
- ∠BOC (circumcentre) = 2A but ∠BIC (incentre) = 90° + A/2 — do not mix them.
📌 3. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. Each interior angle of a regular 20-sided polygon measures: Practice
Correct answer: (B)
Solution: (n − 2) × 180/n = 18 × 180/20 = 162°.
Solution: (n − 2) × 180/n = 18 × 180/20 = 162°.
Q2. Each exterior angle of a regular polygon is 30°. The number of its sides is: Practice
Correct answer: (C)
Solution: n = 360/30 = 12.
Solution: n = 360/30 = 12.
Q3. In △ABC, ∠A = 40°. If O is the circumcentre, ∠BOC = ? Practice
Correct answer: (D)
Solution: ∠BOC = 2∠A = 80°.
Solution: ∠BOC = 2∠A = 80°.
Q4. The two shorter sides of a right triangle are 8 cm and 15 cm. The hypotenuse is: Practice
Correct answer: (A)
Solution: √(8² + 15²) = √289 = 17 cm.
Solution: √(8² + 15²) = √289 = 17 cm.
Q5. Corresponding sides of two similar triangles are in the ratio 1 : 2. The ratio of their areas is: Practice
Correct answer: (B)
Solution: Area ratio = (side ratio)² = 1 : 4.
Solution: Area ratio = (side ratio)² = 1 : 4.
Q6. The radius of a circle is 6 cm and an external point is 10 cm from the centre. The length of the tangent from that point is: Practice
Correct answer: (C)
Solution: Tangent ⟂ radius → √(10² − 6²) = 8 cm.
Solution: Tangent ⟂ radius → √(10² − 6²) = 8 cm.
Q7. In a cyclic quadrilateral ABCD, ∠A = 80°. ∠C = ? Practice
Correct answer: (D)
Solution: Opposite angles sum to 180° → ∠C = 180° − 80° = 100°.
Solution: Opposite angles sum to 180° → ∠C = 180° − 80° = 100°.
Q8. How many diagonals does a polygon with 10 sides have? Practice
Correct answer: (A)
Solution: n(n − 3)/2 = 10×7/2 = 35.
Solution: n(n − 3)/2 = 10×7/2 = 35.
Q9. Two interior angles of a triangle are 64° and 63°. The exterior angle at the third vertex is: Practice
Correct answer: (B)
Solution: Exterior angle = sum of interior opposite angles = 127°.
Solution: Exterior angle = sum of interior opposite angles = 127°.
Q10. Each interior angle of a regular 8-sided polygon measures: Practice
Correct answer: (C)
Solution: (n − 2) × 180/n = 6 × 180/8 = 135°.
Solution: (n − 2) × 180/n = 6 × 180/8 = 135°.
Q11. Each exterior angle of a regular polygon is 72°. The number of its sides is: Practice
Correct answer: (D)
Solution: n = 360/72 = 5.
Solution: n = 360/72 = 5.
Q12. In △ABC, ∠A = 80°. If O is the circumcentre, ∠BOC = ? Practice
Correct answer: (A)
Solution: ∠BOC = 2∠A = 160°.
Solution: ∠BOC = 2∠A = 160°.
Q13. Corresponding sides of two similar triangles are in the ratio 3 : 4. The ratio of their areas is: Practice
Correct answer: (B)
Solution: Area ratio = (side ratio)² = 9 : 16.
Solution: Area ratio = (side ratio)² = 9 : 16.
Q14. The radius of a circle is 9 cm and an external point is 15 cm from the centre. The length of the tangent from that point is: Practice
Correct answer: (C)
Solution: Tangent ⟂ radius → √(15² − 9²) = 12 cm.
Solution: Tangent ⟂ radius → √(15² − 9²) = 12 cm.
Q15. In a cyclic quadrilateral ABCD, ∠A = 85°. ∠C = ? Practice
Correct answer: (D)
Solution: Opposite angles sum to 180° → ∠C = 180° − 85° = 95°.
Solution: Opposite angles sum to 180° → ∠C = 180° − 85° = 95°.
Q16. How many diagonals does a polygon with 12 sides have? Practice
Correct answer: (A)
Solution: n(n − 3)/2 = 12×9/2 = 54.
Solution: n(n − 3)/2 = 12×9/2 = 54.
Q17. Two interior angles of a triangle are 62° and 36°. The exterior angle at the third vertex is: Practice
Correct answer: (B)
Solution: Exterior angle = sum of interior opposite angles = 98°.
Solution: Exterior angle = sum of interior opposite angles = 98°.
Q18. Each interior angle of a regular 10-sided polygon measures: Practice
Correct answer: (C)
Solution: (n − 2) × 180/n = 8 × 180/10 = 144°.
Solution: (n − 2) × 180/n = 8 × 180/10 = 144°.
Q19. The two shorter sides of a right triangle are 14 cm and 48 cm. The hypotenuse is: Practice
Correct answer: (D)
Solution: √(14² + 48²) = √2500 = 50 cm.
Solution: √(14² + 48²) = √2500 = 50 cm.
Q20. Corresponding sides of two similar triangles are in the ratio 2 : 3. The ratio of their areas is: Practice
Correct answer: (A)
Solution: Area ratio = (side ratio)² = 4 : 9.
Solution: Area ratio = (side ratio)² = 4 : 9.