Triangles for Math Olympiad: Complete Preparation Guide

Why This Matters

Triangles are arguably the most important geometric topic in Math Olympiads. Congruence criteria, similarity, the Basic Proportionality Theorem, and Pythagoras theorem — these concepts appear in nearly every competition paper.

For Class 9-10 students, Olympiad triangle problems require combining multiple concepts. A single problem might test congruence, similarity, and Pythagoras — all in one elegant chain of reasoning.

Best Strategy

Common Pitfalls

Mistakes:

• Congruence vs similarity — Congruent = same size and shape. Similar = same shape, possibly different size.
• Corresponding parts — In congruent/similar triangles, match vertices in the correct order.
• BPT direction — BPT applies when a line is parallel to one side. The converse also holds.
• Pythagorean converse — If $a^2 + b^2 = c^2$, the triangle IS right-angled. If $a^2 + b^2 > c^2$, it is acute.

Practice Questions

Try these!

Question 1: Similarity

In triangles ABC and DEF, $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = \frac{2}{3}$. If area of ABC is 36 sq cm, find area of DEF.

Solution: Ratio of areas = square of ratio of sides = $(\frac{2}{3})^2 = \frac{4}{9}$.
Area of DEF = $36 \times \frac{9}{4} = 81$ sq cm.

Question 2: BPT

In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 4, DB = 6, AE = 3, find EC.

Solution: By BPT: $\frac{AD}{DB} = \frac{AE}{EC}$
$\frac{4}{6} = \frac{3}{EC}$
$EC = \frac{6 \times 3}{4} = 4.5$

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