Q.3 (B)(4) Prove that, ‘If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the side in the same proportion’.
Best answer
Answer: Proof by the AA test and the ratio of areas of triangles.
Given : In Δ ABC, line DE || side BC; D lies on AB and E lies on AC (A – D – B, A – E – C).
To prove : \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\)
Construction : Draw seg BE and seg CD.
Proof : Δ ADE and Δ BDE have the same height from E to line AB, so
\(\dfrac{A(\Delta ADE)}{A(\Delta BDE)} = \dfrac{AD}{DB}\) ... (I)
Δ ADE and Δ CDE have the same height from D to line AC, so
\(\dfrac{A(\Delta ADE)}{A(\Delta CDE)} = \dfrac{AE}{EC}\) ... (II)
Δ BDE and Δ CDE have the same base DE and lie between the same parallels DE and BC, so their areas are equal: A(Δ BDE) = A(Δ CDE) ... (III)
From (I), (II) and (III), \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\). Hence proved.
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