Question

In $\Delta$ABC, if DE divides AB and AC in the same ratio, then which of the following options is true?

• A. AD = AE
• B. AD = DB
• C. DE and BC are parallel
• D. DE is half of BC

Answer 1

The correct option is C

DE and BC are parallel

Construction: Draw DE' parallel to BC as shown in the figure above.

Since, DE' II BC,

$\frac{AD}{DB}=\frac{A{E}^{\prime }}{E^{\prime }C}$ ... (i) [by Basic Proportionality Theorem]

But we are given that AD divides AB and AC in the same ratio then

$\frac{AD}{DB}=\frac{AE}{EC}$ ...(ii)

From (i) and (ii), we get

$\frac{A{E}^{\prime }}{E^{\prime }C}=\frac{AE}{EC}$

Adding 1 to both sides further, we get

$\frac{A{E}^{\prime }}{E^{\prime }C}+1=\frac{AE}{EC}+1$

$\frac{AC}{E^{\prime }C}=\frac{AC}{EC}$

$E^{\prime }C=EC$

This is possible only if E' and E coincide. (Since E' and E lie on the same line)

This implies DE' = DE

Hence, DE $\parallel$ BC.