📚 Concepts

📚 Similar Figures

Figures that have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are in the same ratio.

📚 Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are in the same ratio.

📚 Criteria for Similarity of Triangles

🔹 AA (Angle-Angle): Two pairs of corresponding angles are equal.

🔹 SSS (Side-Side-Side): Corresponding sides are in the same ratio.

🔹 SAS (Side-Angle-Side): Two pairs of sides are in the same ratio and the included angle is equal.

📚 Basic Proportionality Theorem (Thales’ Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides those sides in the same ratio.

📚 Criteria for Congruence vs Similarity

Congruent triangles have equal sides and angles, while similar triangles have equal angles but proportional sides.

📚 Areas of Similar Triangles

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

📚 Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

📚 Converse of Pythagoras Theorem

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled.

✍️ Formulas

✍️ Basic Proportionality Theorem

If DE ∥ BC in triangle ABC, and D, E lie on AB and AC respectively, then:

• 𝐀𝐃⁄𝐃𝐁 = 𝐀𝐄⁄𝐄𝐂

✍️ Criteria of Similar Triangles

• AA Similarity → ∠A = ∠D and ∠B = ∠E ⟹ △ABC ∼ △DEF

• SSS Similarity → AB/DE = BC/EF = AC/DF ⟹ △ABC ∼ △DEF

• SAS Similarity → AB/DE = AC/DF and ∠A = ∠D ⟹ △ABC ∼ △DEF

✍️ Area of Similar Triangles

(𝐀𝐫𝐞𝐚 𝐨𝐟 △𝐀𝐁𝐂) ⁄ (𝐀𝐫𝐞𝐚 𝐨𝐟 △𝐃𝐄𝐅) = (𝐀𝐁⁄𝐃𝐄)² = (𝐁𝐂⁄𝐄𝐅)² = (𝐀𝐂⁄𝐃𝐅)²

✍️ Pythagoras Theorem

In △ABC, right-angled at B:

• 𝐀𝐂² = 𝐀𝐁² + 𝐁𝐂²

✍️ Converse of Pythagoras Theorem

If 𝐀𝐂² = 𝐀𝐁² + 𝐁𝐂², then △ABC is right-angled at B.