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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

CAT · 2022 · Quant Slot 2
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

Options

(1) √6
(2) √5
(3) √8
(4) √7
Answer: (4) √7

Explanation:
Let BD=x, then DC=3-x. Area ratio: Area(ADC)/Area(ABD)=DC/BD=1/2, so (3-x)/x=1/2, giving x=2, DC=1. Using coordinate geometry or distance formula with D at distance 2 from B along BC, we can calculate AD=√7.

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