Triangles

State and prove Basic Proportionality Theorem (Thales Theorem). [CBSE 2025]

In △ABC, DE || BC. If AD = 4cm, DB = 6cm and AE = 5cm, find AC. [CBSE 2024]

Prove that if in a triangle, square on one side equals sum of squares on other two sides, angle opposite to first side is right angle. [CBSE 2023]

Two triangles are similar with ratio of corresponding sides 3:5. Find ratio of their areas. [CBSE 2022]

In △ABC, ∠B = 90°, AB = 6cm, BC = 8cm. Find AC. [CBSE 2021]

If a line is drawn parallel to one side of triangle, it divides other two sides in same ratio:\nIf DE || BC, then AD/DB = AE/EC

AAA (Angle-Angle-Angle)\nSAS (Side-Angle-Side)\nSSS (Side-Side-Side)

In right triangle: AC² = AB² + BC²\n(Hypotenuse)² = (Base)² + (Perpendicular)²

Ratio of areas of two similar triangles = Square of ratio of corresponding sides

In △ABC, if DE ∥ BC, then AD/DB = AE/EC. Proof: Draw perpendiculars from D and E to AB. Using areas of triangles ADE, ADB, ADE, AEC and the fact that triangles with equal heights have areas proportional to bases.

In right triangle ABC with ∠B = 90°: AC² = AB² + BC². Proof using areas: Draw squares on all three sides. The area of square on hypotenuse equals sum of areas of squares on other two sides.

Given: In △ABC, DE || BC\nTo Prove: AD/DB = AE/EC\nJoin BE and CD. Draw perpendiculars from E and D to AB.\nUsing ar(△ADE) as base: AD×h₁ and AE×h₂\nSince DE || BC, perpendiculars are proportional\nTherefore AD/DB = AE/EC

Using areas: Draw squares on all three sides of right triangle. Area of square on hypotenuse = Sum of areas of squares on other two sides.