Find the equation of the perpendicular bisector of the line segment joining the points (2, 6) and (4, 0).

Introduction / Context:
This coordinate geometry question asks for the perpendicular bisector of a line segment defined by two points. The perpendicular bisector is the line that passes through the midpoint of the segment and is perpendicular to it. Such questions are common in analytic geometry and test understanding of slope, midpoint formula and basic line equations.

Given Data / Assumptions:

• Endpoints of the segment are A(2, 6) and B(4, 0).
• We need the equation of the perpendicular bisector of segment AB.
• The coordinate system is the standard Cartesian plane.

Concept / Approach:
To find a perpendicular bisector we take two main steps. First, compute the midpoint of the segment using the midpoint formula. Second, determine the slope of the original segment and then find the negative reciprocal to get the slope of the perpendicular bisector. With a point and a slope, we can write the equation of the required line in point slope form and then convert it into one of the standard linear forms to match the options.

Step-by-Step Solution:
Step 1: Compute the midpoint M of A(2, 6) and B(4, 0): M = ((2 + 4)/2, (6 + 0)/2) = (3, 3). Step 2: Find the slope of AB: slope AB = (0 - 6) / (4 - 2) = -6 / 2 = -3. Step 3: The perpendicular bisector has slope m = 1/3, the negative reciprocal of -3. Step 4: Use point slope form with point M(3, 3): y - 3 = (1/3)(x - 3). Step 5: Multiply both sides by 3: 3y - 9 = x - 3. Step 6: Rearrange to standard form: x - 3y = -6.

Verification / Alternative check:
Check that the midpoint M(3, 3) satisfies x - 3y = -6. Substituting gives 3 - 9 = -6, which is true. Also verify that the slopes are perpendicular: slope of AB is -3 and slope of the bisector is 1/3, and their product is -1, confirming perpendicularity.

Why Other Options Are Wrong:

• Option A: x + 3y = 6 does not pass through the midpoint (3, 3).
• Option B: x + 3y = -6 also fails for point (3, 3) since 3 + 9 is not -6.
• Option D: x - 3y = 6 passes through a different region and does not include (3, 3).
• Option E: 3x + y = 12 does not pass through the midpoint and has the wrong slope.

Common Pitfalls:
Students sometimes forget to use the midpoint and instead choose a line simply perpendicular to AB but not passing through the correct point. Another frequent error is in computing the negative reciprocal of the slope; some learners wrongly use the same slope or the reciprocal without changing the sign. Careful computation of both the midpoint and the perpendicular slope avoids these issues.

Final Answer:
Therefore, the equation of the perpendicular bisector is x - 3y = -6.