Question

The equation of the perpendicular bisector of the line segment joining $A(-2, 3)$ and $B(6, -5)$ is

• A. $x + y = 3$
• B. $x + y = 1$
• C. $x - y = -1$
• D. $x - y = 3$

Hint:

Any point on a perpendicular bisector is equidistant from the two endpoints. You can quickly plug the midpoint coordinates $(2, -1)$ directly into the options to find the answer. Checking options:
(A) $2 + (-1) = 1 \ne 3$
(D) $2 - (-1) = 3 = 3$ (Valid!)
This coordinate check can eliminate wrong choices instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Use the geometric property of perpendicular bisectors—equidistance from endpoints—to test the midpoint in given line equations.

Step 2: Key Formula or Approach:
The midpoint of segment $AB$ with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. Any point on the perpendicular bisector satisfies the line equation, particularly the midpoint itself.

Step 3: Detailed Explanation:
Computing the midpoint gives $(2, -1)$. Substituting into option (A): $2 + (-1) = 1 \neq 3$, so it fails. Substituting into option (D): $2 - (-1) = 3 = 3$, which holds true. The other options can be similarly tested and eliminated, leaving (D) as the only equation passing through the midpoint.

Step 4: Final Answer:
The perpendicular bisector equation is the one satisfied by the midpoint $(2, -1)$, matching option (D).