CAT 2022 Slot 2 QA Question 5: The length of each side of an equilateral triangle ABC is 3 cm. Let D be a point on BC such that the area of triangle ADC is half the area of triangle ABD. Then the length of AD, in cm, is

Question

The length of each side of an equilateral triangle ABC is 3 cm. Let D be a point on BC such that the area of triangle ADC is half the area of triangle ABD. Then the length of AD, in cm, is

Accepted Answer

Triangles ADC and ABD have the same altitude from A to BC. Hence, their areas are proportional to their bases DC and DB.

$\dfrac{	ext{Area}(ADC)}{	ext{Area}(ABD)} = \dfrac{DC}{DB} = \dfrac{1}{2}$

Since $BC = 3$, we get $DC = 1$ and $DB = 2$.

Let $M$ be the midpoint of $BC$. In an equilateral tr