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
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
Updated On: Jul 22, 2025
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Correct Answer: 30
Solution and Explanation
Given:
- \( AD = 2x \)
- \( AE = 2y \)
- \( AB = 3x \)
- \( AC = 5y \)
- \( \angle A \) is common to both triangles \( \triangle ADE \) and \( \triangle ABC \)
- Area of \( \triangle ADE = 8 \)
Step 1: Area of \( \triangle ADE \)
Use the formula for area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{included angle}) \] So, \[ \text{Area of } \triangle ADE = \frac{1}{2} \times AD \times AE \times \sin A \] Substituting values: \[ = \frac{1}{2} \times 2x \times 2y \times \sin A = 8 \] \[ \Rightarrow 2x \cdot 2y \cdot \frac{1}{2} \cdot \sin A = 8 \Rightarrow 2xy \cdot \sin A = 8 \Rightarrow xy \cdot \sin A = 4 \tag{1} \]
Step 2: Area of \( \triangle ABC \)
Again using the area formula: \[ \text{Area of } \triangle ABC = \frac{1}{2} \times AB \times AC \times \sin A \] Substituting values: \[ = \frac{1}{2} \times 3x \times 5y \times \sin A = \frac{15}{2} \cdot xy \cdot \sin A \] From equation (1), substitute \( xy \cdot \sin A = 4 \): \[ \text{Area} = \frac{15}{2} \cdot 4 = 30 \]
Final Answer:
∴ Area of triangle \( ABC = 30 \)
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