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

Question

Look the deltoid ABCD shown below.

The ratio between AO and OC is 1:5.

Calculate the ratio between triangle ABD and triangle BCD.

00:00 Calculate the ratio of triangles ABD to BCD

00:03 We'll use the formula to calculate the area of triangle ABD

00:18 We'll do the same thing for triangle BCD

00:30 We'll substitute the triangles' formulas in the ratio

00:41 We'll simplify what we can

00:53 And that's the solution to the question

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 $1:5$ .

1:5