Mathematics — CBSE Class 10 board question

Theorem (Basic Proportionality Theorem / Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Given: In △ABC, DE ∥ BC, intersecting AB at D and AC at E.

To Prove: $\dfrac{AD}{DB} = \dfrac{AE}{EC}$

Construction: Join BE and CD; draw EN ⊥ AB and DM ⊥ AC.

Proof:

$$\frac{\text{ar(ADE)}}{\text{ar(BDE)}} = \frac{\frac{1}{2} \cdot AD \cdot EN}{\frac{1}{2} \cdot DB \cdot EN} = \frac{AD}{DB} \quad \cdots(1)$$

$$\frac{\text{ar(ADE)}}{\text{ar(DEC)}} = \frac{\frac{1}{2} \cdot AE \cdot DM}{\frac{1}{2} \cdot EC \cdot DM} = \frac{AE}{EC} \quad \cdots(2)$$

Since △BDE and △DEC lie on the same base DE and between the same parallels BC and DE:

$$\text{ar(BDE)} = \text{ar(DEC)} \quad \cdots(3)$$

From (1), (2), and (3):

$$\boxed{\frac{AD}{DB} = \frac{AE}{EC}}$$

Converse (Theorem 6.2): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Source: Chapter 6, Section 6.3 – Similarity of Triangles

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• The proof uses area method: express the ratio of areas two ways, then use the fact that triangles on the same base between the same parallels have equal areas.
• Steps (1), (2), (3) and the conclusion must be clearly numbered — examiners award marks for each logical step.
• Always state Given, To Prove, Construction, Proof for a theorem question; missing any costs marks.
• The converse must be stated clearly in one sentence; you do not need to prove it unless the question asks.
• This is a standard 5-mark theorem — the proof is worth ~4 marks and the converse statement ~1 mark.