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In triangle ABC, AD : DB = 2 : 3, DE is parallel to BC.

In triangle ABC, AD : DB = 2 : 3, DE is parallel to BC. Concise Mathematics Solutions ICSE Class 10.

Assertion (A) : DEBC=ADBD=23\dfrac{\text{DE}}{\text{BC}} = \dfrac{\text{AD}}{\text{BD}} = \dfrac{2}{3}.

Reason (R) : DEBC=ADAB=25\dfrac{\text{DE}}{\text{BC}} = \dfrac{\text{AD}}{\text{AB}} = \dfrac{2}{5}.

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true and R is correct reason for A.

  4. Both A and R are true and R is incorrect reason for A.

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Answer

In Δ ADE and Δ ABC

⇒ ∠DAE = ∠BAC (Common angle)

⇒ ∠ADE = ∠ABC (Corresponding angles are equal)

⇒ ∠AED = ∠ACB (Corresponding angles are equal)

∴ Δ ADE ∼ Δ ABC (By AAA postulate)

According to basic proportionality theorem, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

∴DEBC=ADDB⇒DEBC=23\therefore \dfrac{DE}{BC} = \dfrac{AD}{DB} \\[1em] \Rightarrow \dfrac{DE}{BC} = \dfrac{2}{3}

So, assertion is true but reason is false.

Hence, option 1 is the correct option.

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