In $\triangle$ ABC, D is a point on side BC such that $\frac{BD}{DC}=\frac{3}{2}$. If the area of $\triangle$ ABD is 54\ $\text{cm}^2$ answer the following: 1. Find the area of $\triangle$ ACD. 2. Find the area of $\triangle$ ABC. 3. A point E lies on AB such that DE $\parallel AC$. Find the ratio BE : BA. 4. Hence, find the ratio of the areas of $\triangle$ BDE and $\triangle$ BAC.

Step 1: Relate Areas of Triangles with Same Height

When two triangles share the same vertex and their bases lie on the same straight line, their heights are equal. In this case, triangles $ABD$ and $ACD$ share vertex $A$ and their bases $BD$ and $DC$ lie on the line $BC$. Therefore, the ratio of their areas is equal to the ratio of their bases.

Step 2: Calculate Area of Triangle ACD

We are given that the area of $\triangle ABD$ is $54\ \text{cm}^2$ and the ratio $BD:DC = 3:2$. By substituting these values into the area ratio formula, we can calculate the area of $\triangle ACD$.

Step 3: Calculate Area of Triangle ABC

The area of $\triangle ABC$ is the sum of the areas of $\triangle ABD$ and $\triangle ACD$, as point $D$ lies on side $BC$.

Step 4: Find Ratio BE : BA using Thales Theorem

Since $DE \parallel AC$, by the Basic Proportionality Theorem (also known as Thales Theorem), the line $DE$ divides the sides $AB$ and $BC$ in the same ratio. This means $BE/EA = BD/DC$. From this, we can find the ratio $BE:BA$.

Step 5: Find Ratio of Areas of Triangles BDE and BAC

Since $DE \parallel AC$, $\triangle BDE$ is similar to $\triangle BAC$. For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. We found $BE:BA = 3:5$, so we can use this ratio.

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