CAT 2022 Geometry & Mensuration — Previous Year Question

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