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Mathematics

Triangles

Introduction

This chapter deals with the similarity of triangles. Two figures having the same shape but not necessarily the same size are called similar figures. You will learn criteria for similarity of triangles (AA, SSS, SAS), and important theorems like the Basic Proportionality Theorem (BPT). The Pythagoras Theorem and its converse are also covered, which are fundamental to coordinate geometry and trigonometry.

1

Basic Proportionality Theorem (BPT / Thales Theorem)

The Basic Proportionality Theorem states: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. That is, in triangle ABC, if DE is parallel to BC (where D is on AB and E is on AC), then AD/DB = AE/EC. The converse is also true: if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Key Points

  • •If DE || BC in triangle ABC, then AD/DB = AE/EC
  • •Equivalently: AD/AB = AE/AC = DE/BC
  • •The converse is also true and is used to prove lines are parallel
  • •BPT is used as a stepping stone in many similarity proofs
  • •Also known as Thales' theorem

Worked Example

In triangle ABC, DE || BC, AD = 3 cm, DB = 5 cm, AE = 4.5 cm. Find EC. By BPT: AD/DB = AE/EC 3/5 = 4.5/EC EC = 4.5 x 5/3 = 7.5 cm

Watch Out

When using BPT, be careful about which ratios to equate. AD/DB = AE/EC (not AD/AB = AE/EC, unless you use that form consistently).

2

Criteria for Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. There are three criteria to establish similarity without checking all conditions: (1) AA (Angle-Angle): if two angles of one triangle are equal to two angles of another; (2) SSS (Side-Side-Side): if all three pairs of corresponding sides are proportional; (3) SAS (Side-Angle-Side): if one pair of corresponding sides are proportional and the included angles are equal.

Key Points

  • •AA criterion: two pairs of equal angles imply similarity
  • •SSS criterion: all three pairs of sides in the same ratio
  • •SAS criterion: two pairs of sides in same ratio with included angle equal
  • •Always write similarity in correct order: triangle ABC ~ triangle DEF means angle A = angle D, etc.
  • •The ratio of corresponding sides is called the scale factor

Worked Example

In triangles ABC and DEF, angle A = angle D = 50 degrees and angle B = angle E = 70 degrees. Are they similar? Yes, by AA criterion, triangle ABC ~ triangle DEF. Therefore AB/DE = BC/EF = AC/DF.

Watch Out

The ORDER of vertices in the similarity statement is crucial. If triangle ABC ~ triangle PQR, then A corresponds to P, B to Q, C to R.

3

Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If triangle ABC ~ triangle DEF with scale factor k (i.e., AB/DE = k), then Area(ABC)/Area(DEF) = k^2. This extends to all corresponding linear measurements: medians, altitudes, and perimeters are in ratio k, while areas are in ratio k^2.

Key Points

  • •Area ratio = (side ratio)^2 for similar triangles
  • •If sides are in ratio k, areas are in ratio k^2
  • •Perimeters of similar triangles are in the same ratio as corresponding sides
  • •Medians, altitudes, angle bisectors are also in the ratio of corresponding sides
  • •This theorem is very useful in finding areas when similarity is established

Worked Example

Triangle ABC ~ triangle DEF and AB/DE = 3/4. If area of triangle DEF = 64 sq cm, find area of triangle ABC. Area(ABC)/Area(DEF) = (AB/DE)^2 = (3/4)^2 = 9/16 Area(ABC) = 64 x 9/16 = 36 sq cm

4

Pythagoras Theorem and Its Converse

The Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If triangle ABC is right-angled at B, then AC^2 = AB^2 + BC^2. The converse states: if in a triangle, the square of one side equals the sum of squares of the other two sides, then the angle opposite to the first side is a right angle.

Key Points

  • •In right triangle: (hypotenuse)^2 = (base)^2 + (perpendicular)^2
  • •The hypotenuse is always the longest side, opposite the right angle
  • •Converse: if c^2 = a^2 + b^2, the triangle is right-angled
  • •Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
  • •Used extensively in coordinate geometry and trigonometry

Worked Example

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. Let the distance be x metres. By Pythagoras: 10^2 = 8^2 + x^2 100 = 64 + x^2 x^2 = 36 x = 6 m

Watch Out

In board exams, proofs of BPT and Pythagoras theorem are common 5-mark questions. While proofs of the area theorem and Pythagoras theorem are deleted, their applications are still in syllabus.

Quick Summary

  • ✓BPT: line parallel to one side divides other two sides proportionally
  • ✓AA, SSS, SAS are the three criteria for similarity
  • ✓Order of vertices in similarity statement matters
  • ✓Area ratio of similar triangles = (side ratio)^2
  • ✓Pythagoras: hypotenuse^2 = base^2 + height^2
  • ✓Converse of Pythagoras: used to prove a triangle is right-angled
  • ✓All corresponding linear measurements of similar triangles are in ratio k
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Key Formulas

BPT: If DE || BC in triangle ABC, AD/DB = AE/EC

Area(triangle1)/Area(triangle2) = (side1/side2)^2

Pythagoras: AC^2 = AB^2 + BC^2 (right-angled at B)

Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25)

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