Calculate Triangle Ratios in a Deltoid: Using the 1:5 Point Relationship

To determine the area ratio between triangles $\triangle ABD$ and $\triangle BCD$, we will compare the segments derived from the given point O.

• We are given that the ratio $AO : OC = 1 : 5$.
• Both triangles $\triangle ABD$ and $\triangle BCD$ share the line segment BD as a base, with their 'perpendicular heights' being the same when considering B as the vertex and segments AO and OC as part of their respective triangles.
• The areas of triangles sharing the same base and height are proportional to the lengths of the other segment partitions they connect to.
• Since O divides AC in the mentioned ratio, AO is $1/6$ of AC and OC is $5/6$ of AC.

Thus, the ratio of the areas of the triangles, based on the aforementioned proportions of their respective line segments, becomes $\frac{1}{5}$.

This simplifies to the answer of $1:5$

Therefore, the ratio of the areas of triangle $\triangle ABD$ to triangle $\triangle BCD$ is $1:5$.