OFFICIAL EXPLANATION

Let AE be the perpendicular (Altitude) for both the triangles $\(\mathrm{ABD} \& \mathrm{BDC}\)$

As triangle BDC has all angles equal, it is an equilateral triangle, let each of its side be ' $\(n\)$ '
Next triangle $\(A B D\)$ is an isosceles triangle as angle $\(D A B\)$ is same as angle $\(D B A\)$, so we get $\(A D=\)$ $\(B D=n\)$ only (as $\(B D\)$ is also one of the sides of the equilateral triangle $\(A D C\)$ )

Thus, both the triangles $\(\mathrm{ABD} \& \mathrm{BDC}\)$ have equal altitude ( BE each) and equal length bases ( n each), it implies both the triangles are equal in area.

Hence column $\(A\)$ has same quantity as column $B$, so the answer is (C).