Triangles

What this chapter is about

This chapter explores the concept of similarity of triangles, building upon the idea of congruence studied in earlier classes. While congruent figures are exact copies (same shape and size), similar figures have the same shape but may differ in size. You will learn precise conditions under which two triangles are similar and how their sides and areas relate.

The chapter establishes criteria for similarity of triangles, analogous to the congruence criteria (SAS, ASA, SSS, RHS) you already know. These similarity criteria—AAA, AA, SSS and SAS—allow you to prove triangles similar without checking every angle and every side ratio. You will also study important theorems about lines parallel to a side of a triangle and the relationship between areas of similar triangles.

After studying this chapter, you should be able to identify similar triangles, apply similarity criteria in proofs and problems, use the Basic Proportionality Theorem, and calculate unknown lengths and areas using ratios derived from similarity.

Key ideas

• Two polygons are similar if their corresponding angles are equal and corresponding sides are in the same ratio (called the scale factor or ratio of similitude).
• Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• The converse also holds: If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
• AAA (or AA) Criterion: If in two triangles, all three pairs (or any two pairs) of corresponding angles are equal, the triangles are similar.
• SSS Criterion for Similarity: If the corresponding sides of two triangles are in the same ratio, the triangles are similar.
• SAS Criterion for Similarity: If one angle of a triangle equals one angle of another, and the sides including these angles are in the same ratio, the triangles are similar.
• The ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides.
• In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse divides the triangle into two triangles, each similar to the original and to each other.

Formulas and facts to remember

• If △ABC ~ △DEF with scale factor k, then AB/DE = BC/EF = CA/FD = k.
• Area(△ABC) / Area(△DEF) = k² = (AB/DE)² = (BC/EF)² = (CA/FD)².
• Basic Proportionality Theorem: If DE ∥ BC in △ABC with D on AB and E on AC, then AD/DB = AE/EC.
• Pythagoras Theorem (special case of similarity): In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
• Converse of Pythagoras Theorem: If in a triangle the square of one side equals the sum of the squares of the other two sides, the angle opposite the first side is a right angle.

Worked examples

Example 1: Using the Basic Proportionality Theorem

In △PQR, a line parallel to QR meets PQ at X and PR at Y. If PX = 4 cm, XQ = 6 cm and PY = 5 cm, find YR.

Solution: Since XY ∥ QR, by the Basic Proportionality Theorem: PX/XQ = PY/YR 4/6 = 5/YR YR = 5 × 6/4 = 30/4 = 7.5 cm

Example 2: Proving triangles similar and finding an unknown side

In △ABC and △PQR, angle A = 50°, angle B = 60°, angle P = 50° and angle Q = 60°. If AB = 6 cm, BC = 8 cm and PQ = 9 cm, find QR.

Solution: First, check similarity. Angle A = angle P = 50° Angle B = angle Q = 60° Therefore angle C = angle R = 180° − 50° − 60° = 70°

By the AA criterion, △ABC ~ △PQR.

Corresponding sides are proportional: AB/PQ = BC/QR 6/9 = 8/QR QR = 8 × 9/6 = 72/6 = 12 cm

Example 3: Ratio of areas of similar triangles

Two similar triangles have corresponding sides in the ratio 3 : 5. If the area of the smaller triangle is 27 cm², find the area of the larger triangle.

Solution: Let the ratio of corresponding sides be 3/5. Ratio of areas = (ratio of sides)² = (3/5)² = 9/25

Let area of larger triangle be A. 27/A = 9/25 A = 27 × 25/9 = 675/9 = 75 cm²

Common mistakes

• Confusing congruence with similarity → Congruent triangles are equal in size; similar triangles only have equal angles and proportional sides.
• Writing ratios of non-corresponding sides → Always match vertices in the same order; if △ABC ~ △DEF, then AB corresponds to DE, not DF.
• Forgetting to square the ratio when comparing areas → Area ratio is the square of the side ratio, not the same as the side ratio.
• Assuming any two triangles with proportional sides are congruent → They are similar, but congruent only when the scale factor is 1.
• Applying the Basic Proportionality Theorem when the line is not parallel to a side → The theorem requires the line to be parallel to one side of the triangle.

Quick revision

• Similar triangles have equal corresponding angles and proportional corresponding sides.
• A line parallel to one side of a triangle divides the other two sides proportionally (BPT).
• Two triangles are similar if two pairs of angles are equal (AA criterion).
• Ratio of areas of similar triangles = (ratio of corresponding sides)².
• The altitude to the hypotenuse of a right triangle creates two triangles similar to the original.
• Pythagoras Theorem follows from similarity of right triangles.