How to Find the Gradient of the Perpendicular Bisector?

To find the gradient of the perpendicular bisector of a line segment, you need to first determine the gradient of the original line segment and then calculate its negative reciprocal.

Here's a breakdown of the steps:

1. Find the Gradient of the Original Line Segment:

   • If you are given two points, say (x₁, y₁) and (x₂, y₂), the gradient (m) of the line segment connecting them is calculated as:

     m = (y₂ - y₁) / (x₂ - x₁)

   • If you are given the equation of the line in the form y = mx + c, then the gradient is simply the coefficient 'm' of x.

2. Calculate the Gradient of the Perpendicular Bisector:

   • The perpendicular bisector is, by definition, perpendicular to the original line segment. The gradients of perpendicular lines are negative reciprocals of each other.

   • If the gradient of the original line segment is 'm', then the gradient of the perpendicular bisector (m_perp) is:

     m_perp = -1/m

Example:

Let's say you have a line segment with endpoints (1, 2) and (4, 8).

1. Gradient of the line segment:

   m = (8 - 2) / (4 - 1) = 6 / 3 = 2

2. Gradient of the perpendicular bisector:

   m_perp = -1 / 2 = -1/2

Therefore, the gradient of the perpendicular bisector is -1/2.

Summary:

To find the gradient of the perpendicular bisector:

• Calculate the gradient of the original line segment.
• Take the negative reciprocal of that gradient.