In coordinate geometry, what is the area (in square units) of the triangle formed by the x-axis, the y-axis, and the line x + 2y = 6?

Introduction / Context:
This problem uses intercepts of a straight line to find the area of a triangle formed with the coordinate axes. Such questions are standard in coordinate geometry and test understanding of intercept form and basic area formulas for right triangles.

Given Data / Assumptions:

• The line is x + 2y = 6.
• The triangle is formed by this line and the x axis and y axis.
• We need the area in square units.
• The coordinate axes intersect at the origin (0, 0).

Concept / Approach:
Where the line meets the x axis, y = 0. Where it meets the y axis, x = 0. These intercepts provide the base and height of a right triangle with the axes. The area of a right triangle is half the product of its perpendicular sides. So we find the intercepts and apply the formula (1/2) * base * height.

Step-by-Step Solution:
To find the x intercept, set y = 0 in x + 2y = 6. This gives x + 2 * 0 = 6, so x = 6. Thus the x intercept is (6, 0). To find the y intercept, set x = 0 in x + 2y = 6. This gives 0 + 2y = 6, so y = 3. Thus the y intercept is (0, 3). The triangle is formed by points (0, 0), (6, 0), and (0, 3). The base along the x axis has length 6 units. The height along the y axis has length 3 units. Area of the right triangle = (1/2) * base * height = (1/2) * 6 * 3 = 9 square units.

Verification / Alternative check:
We can use the formula for a triangle with intercepts a and b on the axes, where the line is x/a + y/b = 1. For x + 2y = 6, dividing through by 6 gives x/6 + y/3 = 1, so a = 6 and b = 3. The area of the triangle formed with the axes is (1/2) * a * b = (1/2) * 6 * 3 = 9. This matches the earlier calculation, confirming the answer.

Why Other Options Are Wrong:
Values such as 3 or 6 correspond to using only one of the intercepts or forgetting the factor 1/2. An area of 12 would be obtained if someone multiplies 6 and 3 directly without halving. The value 18 is twice the correct area, as if one accidentally doubled instead of halved. Only 9 square units is consistent with the correct formula for the area of a right triangle with legs 6 and 3.

Common Pitfalls:
A common mistake is to misidentify the intercepts if the equation is not first rewritten clearly, or to treat the intercepts as coordinates in a more complicated area formula. For right triangles aligned with the axes, it is much simpler to use (1/2) * base * height directly. Always check that the line indeed intersects both axes in the first quadrant for a positive area interpretation.

Final Answer:
The area of the triangle is 9 square units.