Question

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:

• A. 6abc
• B. 9abc
• C. 6bc
• D. 9ac
• E. None of the above

Answer 1

The Correct Option is D

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