Question

What is the area of a triangle with vertices at (0,0), (3,0), and (0,4)? 
 

• A. 6
• B. 8
• C. 10
• D. 12

Hint:

If a triangle has vertices on axes, the area can be found quickly using $\frac{1}{2} \times$ base $\times$ height.

Answer 1

The Correct Option is A

- Step 1: Recognize the triangle type - The points form a right triangle: 
Base = segment from (0,0) to (3,0) - length 3. 
Height = segment from (0,0) to (0,4) - length 4. 

- Step 2: Area formula for a triangle - \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] 
- Step 3: Substitute values - \[ \text{Area} = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \] 
- Step 4: Alternate check using determinant formula - \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] Substitute $(0,0), (3,0), (0,4)$: \[ = \frac{1}{2} | 0(0-4) + 3(4-0) + 0(0-0) | = \frac{1}{2} | 0 + 12 + 0 | = 6 \] 
- Step 5: Conclusion - Area is $6$, matching option (1).