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
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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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
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