How to Construct a Perpendicular Bisector: Step-by-Step Guide and Equation

Perpendicular Bisector

A perpendicular bisector is a line or a line segment that divides another line segment into two equal parts and also forms a right angle (90 degrees) with it

A perpendicular bisector is a line or a line segment that divides another line segment into two equal parts and also forms a right angle (90 degrees) with it.

To better understand the concept, let’s consider an example. Suppose we have a line segment AB. To create its perpendicular bisector, we need to follow these steps:

1. Find the midpoint: Determine the midpoint of the line segment AB. The midpoint is the point that is equidistant from both A and B. Use the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

For example, if A has coordinates (1, 3) and B has coordinates (5, 7), then the midpoint is:

Midpoint = ((1 + 5)/2, (3 + 7)/2) = (3, 5)

2. Determine the slope: Calculate the slope of the line segment AB using the slope formula:

Slope = (y2 – y1)/(x2 – x1)

Using the coordinates of A and B from the previous example, the slope is:

Slope = (7 – 3)/(5 – 1) = 4/4 = 1

3. Calculate the negative reciprocal: Find the negative reciprocal of the slope calculated in the previous step. To do this, simply flip the fraction and change its sign. For example, if the slope is 1, the negative reciprocal would be -1.

4. Create the equation: Using the negative reciprocal of the slope and the midpoint, create an equation for the line of the perpendicular bisector.

If the slope is -1 and the midpoint is (3, 5), the equation for the perpendicular bisector can be written in point-slope form as:

y – y1 = m(x – x1) where m is the slope and (x1, y1) is the midpoint.

Substitute the values:

y – 5 = -1(x – 3)

Simplify:

y – 5 = -x + 3

y = -x + 8

5. Draw the perpendicular bisector: Plot the midpoint on a coordinate plane. From that point, use the equation to draw the line with the slope of the perpendicular bisector passing through the midpoint. This line will split the original line segment AB into two equal parts at a right angle.

So, the perpendicular bisector of the line segment AB can be represented by the equation y = -x + 8. Any point on this line will be equidistant from both points A and B, and it will form a right angle with the line segment AB.

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