ABC is a triangle and the coordinates of A, B, and C are (a, b - 2c), (a, b + 4c), and (-2a, 3c), respectively, where a, b, and c are positive numbers. The area of the triangle ABC is:
- 6abc
- 9abc
- 6bc
- 9ac
- None of the above
The Correct Option is D
Solution and Explanation
Step 1: Formula for the area of a triangle using coordinates:
\[ \text{Area} = \tfrac{1}{2} \Big| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \Big| \]
Step 2: Substitute the given points into the formula.
\[ \text{Area} = \tfrac{1}{2} \Big| a\big((b+4c) - 3c\big) \;+\; a\big(3c - (b - 2c)\big) \;+\; 2a\big((b - 2c) - (b + 4c)\big) \Big| \]
Step 3: Simplify each term.
\[ = \tfrac{1}{2} \Big| a(b + c) \;+\; a(5c - b) \;-\; 2a(-6c) \Big| \]
Step 4: Expand further.
\[ = \tfrac{1}{2} \Big| ab + ac + 5ac - ab + 12ac \Big| \]
Step 5: Combine like terms.
\[ = \tfrac{1}{2} \Big| 18ac \Big| \]
Step 6: Final area of the triangle.
\[ \text{Area} = 9ac \]
Final Answer:
\[ \boxed{9ac} \]
Correct Option:
Option D
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