Question

If \( A(a,0) \), \( B(0,0) \), and \( C(0,b) \) are the vertices of \( \triangle ABC \), then the area of \( \triangle ABC \) is:

• A. \( ab \)
• B. \( \frac{1}{2} ab \)
• C. \( \frac{1}{2} a^2 b^2 \)
• D. \( \frac{1}{2} b^2 \)

Hint:

Use determinant-based area formula for triangle area calculation.

Answer 1

The Correct Option is B

Step 1: Use the area formula for a triangle The area of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Step 2: Substitute values Given points: \( A(a,0) \), \( B(0,0) \), \( C(0,b) \): \[ \text{Area} = \frac{1}{2} \left| a(0 - b) + 0(b - 0) + 0(0 - 0) \right| \] \[ = \frac{1}{2} \left| -ab \right| \] \[ = \frac{1}{2} ab \] Thus, the correct answer is \( \frac{1}{2} ab \).