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Explanation:

Area of ∆ACD is half of area of ∆ABD.
Since their height is same, ratio of their areas will be same as the ratio of their bases.
⇒ BD = 2CD
⇒ BD = 2 cm and CD = 1 cm.

Taking E as the midpoint of BC, BE = 3/2 = 1.5
⇒ ED = 2 – 1.5 = 0.5 cm.

Also, AE = height of the equilateral triangle = $\frac{\sqrt{3}}{2}$ × 3 = $\frac{3\sqrt{3}}{2}$

In ∆AED
⇒ AD² = AE² + ED²
⇒ AD² = ${\left(\frac{3\sqrt{3}}{2}\right)}^2$ + (0.5)²

⇒ AD² = $\frac{27}{4}$ + $\frac{1}{4}$ = 7

⇒ AD = $\sqrt{7}$

Hence, option (a).