Understanding Triangle Similarity

Given that DE is parallel to BC (DE || BC) in $\Delta$ABC, it implies that $\Delta$ADE is similar to $\Delta$ABC ($\Delta$ADE ~ $\Delta$ABC).

Determining Side Ratios

• The ratio AD : BD is given as 2 : 3.
• This means the ratio of AD to the whole side AB is AD : AB = AD : (AD + BD) = 2 : (2 + 3) = 2 : 5.
• So, the ratio of corresponding sides of the similar triangles is $\frac{AD}{AB} = \frac{2}{5}$.

Relating Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$ \frac{\text{Area}(\Delta \text{ADE})}{\text{Area}(\Delta \text{ABC})} = \left(\frac{AD}{AB}\right)^2 $

$ \frac{\text{Area}(\Delta \text{ADE})}{\text{Area}(\Delta \text{ABC})} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} $

Let the Area($\Delta$ADE) be $4x$. Then, the Area($\Delta$ABC) must be $25x$.

Calculating Trapezium Area

The area of the trapezium BDEC is the difference between the area of the larger triangle $\Delta$ABC and the smaller triangle $\Delta$ADE.

Area(BDEC) = Area($\Delta$ABC) - Area($\Delta$ADE)

Area(BDEC) = $25x - 4x = 21x$.

We are given that the Area(BDEC) = $63 \text{ cm}^2$.

Therefore, $21x = 63 \text{ cm}^2$.

Finding the Area of $\Delta$ADE

Solve for $x$: $ x = \frac{63 \text{ cm}^2}{21} = 3 \text{ cm}^2 $ Now, calculate the Area($\Delta$ADE): $ \text{Area}(\Delta \text{ADE}) = 4x = 4 \times (3 \text{ cm}^2) = 12 \text{ cm}^2 $