Q.Find the equation of the right bisector of the line segment joining the points $(3, 4)$ and $(-1, 2)$.

Concept: Section Formula & Perpendicular Bisector

The right bisector (perpendicular bisector) of a segment is the line that passes through its midpoint and is perpendicular to it.

Step 1 – Midpoint

Midpoint $M$ of $(3,4)$ and $(-1,2)$:

Step 2 – Slope of segment

Slope $m_{AB} = \frac{2 - 4}{-1 - 3} = \frac{-2}{-4} = \frac{1}{2}$

Step 3 – Slope of perpendicular …

The right bisector of a line segment is also called the perpendicular bisector. It is the line that is perpendicular to the segment and passes through its midpoint. Every point on this line is equidistant from the two endpoints — that’s the geometric property that defines it.

To find its equation, we need two things:

1. The midpoint of the segment (where the bisector passes through).
2. The slope of the segment (so we can find the slope of a line perpendicular to it).

Step-by-step solution

1. Find the midpoint of the segment.

The midpoint $M$ of points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Here, $(x_1, y_1) = (3, 4)$ and $(x_2, y_2) = (-1, 2)$.

So:

The right bisector passes through $(1, 3)$.

2. Find the slope of the given segment.

Slope $m$ of the line through $(3,4)$ and $(-1,2)$ is:

3. Find the slope of the perpendicular bisector. …