How Do You Find the Equation of a Perpendicular Bisector Between Two Points?

To find the equation of a perpendicular bisector, calculate the midpoint of the segment, find the segment’s gradient, and use its negative reciprocal as the perpendicular gradient. Then substitute the midpoint and perpendicular gradient into point-gradient form.

This explanation assumes IB Mathematics: Applications and Interpretation SL and a Paper 1-style short response.

For points $A(x_1,y_1)$ and $B(x_2,y_2)$, first calculate the midpoint:

$$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

Next, calculate the gradient of the segment $AB$:

$$m_{AB}=\frac{y_2-y_1}{x_2-x_1}.$$

For non-horizontal and non-vertical lines, perpendicular gradients satisfy

$$m_{AB}m_{\perp}=-1,$$

so the required gradient is

$$m_{\perp}=-\frac{1}{m_{AB}}.$$

Finally, use the point-gradient form through the midpoint $(x_M,y_M)$:

$$y-y_M=m_{\perp}(x-x_M).$$

Worked Example

Find the perpendicular bisector of the segment joining $A(2,3)$ and $B(8,7)$.

The midpoint is

$$M=\left(\frac{2+8}{2},\frac{3+7}{2}\right)=(5,5).$$

The gradient of $AB$ is

$$m_{AB}=\frac{7-3}{8-2}=\frac{2}{3}.$$

Therefore, the perpendicular gradient is $-\frac{3}{2}$. Using the midpoint gives

$$y-5=-\frac{3}{2}(x-5).$$

An equivalent simplified equation is

$$3x+2y=25.$$

To verify the result, substitute $(5,5)$ into the equation and confirm it lies on the bisector. Also check that $\frac{2}{3}\times-\frac{3}{2}=-1$, proving the lines are perpendicular.

Geometrically, every point on a perpendicular bisector is equidistant from the segment’s endpoints. Therefore, as an additional check, choose any point on your final line and confirm that its distances from $A$ and $B$ are equal.

These special cases matter because vertical lines have undefined gradients. Do not attempt to calculate a negative reciprocal when the original segment is horizontal or vertical; identify the bisector directly from the midpoint instead.

A common misconception is to use only the negative gradient, (-m). The correct perpendicular gradient is the negative reciprocal, $-\frac{1}{m}$; both changing the sign and taking the reciprocal are required.

Exam technique: This is SL 3.5: equations of perpendicular bisectors. Show the midpoint, the original gradient, the perpendicular gradient, and the final equation. These are the method steps an examiner expects, even when a GDC is available.