Question

Let D and E be points on sides AB and AC, respectively, of a triangle ABC, such that AD : BD = 2 : 1 and AE : CE = 2 : 3. If the area of the triangle ADE is 8 sq cm, then the area of the triangle ABC, in sq cm, is

Answer 1

Given:

The area of triangle \( \triangle ADE \) is given by:

\[ \text{Area}_{\triangle ADE} = \frac{1}{2} \times AD \times AE \times \sin A \]

Let \( AD = 2x \) and \( AE = 2y \).
Thus: \[ \text{Area}_{\triangle ADE} = \frac{1}{2} \times 2x \times 2y \times \sin A = 8 \]

Simplifying: \[ \Rightarrow 2x \cdot 2y = 4xy \Rightarrow 4xy \cdot \sin A = 8 \Rightarrow xy \cdot \sin A = 2 \]

Calculate the Area of Triangle \( \triangle ABC \):

Given: \[ AB = AD + DB = 2x + x = 3x \] and \[ AC = AE + EC = 2y + 3y = 5y \].
The area of triangle \( \triangle ABC \) is calculated as:

\[ \text{Area}_{\triangle ABC} = \frac{1}{2} \times AB \times AC \times \sin A \]

\[ = \frac{1}{2} \times 3x \times 5y \times \sin A = \frac{15}{2} \cdot xy \cdot \sin A \]

Using the previously derived value \[ xy \cdot \sin A = 2 \]:

\[ \text{Area}_{\triangle ABC} = \frac{15}{2} \cdot 2 = \boxed{15} \]

✅ Final Answer: Area of triangle ABC is 30 cm²