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:
We need to find the line equation that cuts perfectly through the midpoint of the segment joining points $A$ and $B$ at a right angle ($90^\circ$).

Step 2: Key Formula or Approach:
1. Find the coordinates of the midpoint $M$ using the midpoint formula: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ 2. Find the slope of segment $AB$: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$. 3. Calculate the perpendicular slope: $m_{\perp} = -\frac{1}{m_{AB}}$. 4. Substitute the point $M$ and slope $m_{\perp}$ into the point-slope formula: $y - y_1 = m(x - x_1)$.

Step 3: Detailed Explanation:
Let's first calculate the midpoint $M$ of the segment $AB$: $$M = \left(\frac{-2 + 6}{2}, \frac{3 + (-5)}{2}\right) = \left(\frac{4}{2}, \frac{-2}{2}\right) = (2, -1)$$ Now, let's find the slope of line segment $AB$: $$m_{AB} = \frac{-5 - 3}{6 - (-2)} = \frac{-8}{8} = -1$$ Since the bisector line is perpendicular to $AB$, its slope must be the negative reciprocal: $$m_{\perp} = -\frac{1}{-1} = 1$$ Using the point-slope form with slope $m = 1$ passing through the midpoint $(2, -1)$: $$y - (-1) = 1(x - 2)$$ $$y + 1 = x - 2$$ Rearranging into standard form: $$x - y = 3$$

Step 4: Final Answer:
The equation of the perpendicular bisector is $x - y = 3$, which matches option (D).