Triangles — Class 9 (Interactive, English)

Complete formulas, 5 solved problems with diagrams, and 5 practice questions. Interactive SVGs — hover/tap to highlight.

Key Concepts & Definitions

  • Triangle: Polygon with three sides and three angles.
  • Types by sides: Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides).
  • Types by angles: Acute (all <90°), Right (one =90°), Obtuse (one >90°).
  • Interior angles sum to 180°; Exterior angle = sum of two opposite interior angles.
  • Median, Altitude, Perpendicular bisector, Angle bisector — special lines in triangle.
Important formulas
Angle sum: ∠A + ∠B + ∠C = 180°
Exterior angle: ∠Exterior = ∠B + ∠C
Area (base b, height h): Δ = 1/2 × b × h
Heron's formula: s = (a+b+c)/2, Area = √[s(s−a)(s−b)(s−c)]
Pythagoras (right triangle): a² + b² = c²

Interactive Diagrams

Hover over lines/angles/points to highlight. Tap on mobile.

A B C Median from C ∠A ∠B ∠C A B C Right triangle: AC ⟂ AB; a² + b² = c²

5 Solved Problems (with diagrams & steps)

Q1. In triangle ABC, AB = AC. If ∠B = 40°, find ∠A and ∠C.
Isosceles: AB = AC ⇒ base angles at B and C equal. Given ∠B = 40°, so ∠C = 40°. Sum: ∠A + 40 + 40 = 180 ⇒ ∠A = 100°.
Q2. Right triangle with legs 6 cm and 8 cm. Find hypotenuse and area.
Pythagoras: c = √(6² + 8²) = √(36+64) = √100 = 10 cm. Area = 1/2 × 6 × 8 = 24 cm².
Q3. In ΔABC, sides a=13, b=14, c=15. Find area using Heron’s formula.
s=(13+14+15)/2=21. Area = √[21(21−13)(21−14)(21−15)] = √[21×8×7×6] = √(7056) = 84.
Q4. In triangle ABC, AD is median to BC. If AB = 10, AC = 8 and BC = 12, find length of median AD? (Use Apollonius theorem)
Apollonius: AB² + AC² = 2(AD² + (BC²)/4). So 10² + 8² = 2(AD² + 12²/4) ⇒ 100+64=2(AD² +36) ⇒164=2AD² +72 ⇒2AD²=92 ⇒AD²=46 ⇒AD=√46 ≈ 6.782.
Q5. Exterior angle at A is 120° and one interior opposite angle B is 40°. Find C.
Exterior at A = B + C ⇒ 120 = 40 + C ⇒ C = 80°. Check interior sum: A + 40 + 80 = 180 ⇒ A = 60°.

5 Practice Questions

1. Triangle with angles in ratio 2:3:4. Find each angle.
2. In right triangle, hypotenuse = 13 and one leg = 5. Find the other leg and area.
3. Find area of triangle with sides 7, 24, 25.
4. In isosceles triangle, vertex angle = 20°; find base angles.
5. In ΔABC, AB = 14, AC = 10, BC = 12. Find median from A using Apollonius theorem.

Answers (brief)

  1. Angles = 40°, 60°, 80°? Wait compute: ratio 2:3:4 sum 9 parts → total 180 ⇒ each part = 20 ⇒ angles 40°, 60°, 80°.
  2. Other leg = √(13² − 5²) = √(169−25)=√144=12. Area = 1/2 ×5×12 =30.
  3. s=(7+24+25)/2=28 → Area = √[28(21)(4)(3)] = √(7056)=84.
  4. Base angles = (180−20)/2 = 80° each.
  5. Use Apollonius: AB²+AC² = 2(AM² + (BC²)/4) ⇒ 14²+10² = 2(AM² + 36) ⇒196+100=2AM²+72 ⇒296=2AM²+72 ⇒2AM²=224 ⇒AM²=112 ⇒AM=√112=4√7 ≈10.583.