Perpendicular Bisector: Definition, Theorem, Equation, And How To Construct

5 min readOct 17, 2021

A perpendicular bisector is a line that crosses another line at a 90-degree angle, or it is the split of anything into two equal or congruent halves. It might be a line, ray, or segment that divides another line segment into two equal halves at a 90-degree angle.

What Is A Perpendicular Bisector: Perpendicular Bisector Definition Geometry

What is a perpendicular bisector? How can an archaeologist estimate the size of a plate if just a portion of it has been discovered? How does a landscaper evaluate sprinkler placement for the most efficient use of water?

Perpendicular bisector definition: A bisector is anything that divides an item into two equal pieces. Typically, this “something” is a line or a segment. The definition of a perpendicular bisector is that it aids in the determination of the midpoint of a line.

Equation For Perpendicular Bisector

Let’s look at the definition of perpendicular bisector from an equation standpoint. Here is how to solve for the perpendicular bisector equation:

Step 1: Find the mid-point between the two points.

To calculate the midpoint of two points, use the midpoint formula: [(x1 + x2)/2, (y1 + y2)/2]. This implies you’re simply taking the average of the two sets of points’ x and y coordinates, which brings you to the midpoint of the two coordinates. Assume we have the (x1, y1) coordinates of (2, 5) and the (x2, y2) coordinates of (8, 3). Here’s how to determine the intersection of those two points:

[(2+8)/2, (5 +3)/2] (5, 4) = (10/2, 8/2)

The midpoint coordinates of (2, 5) and (8, 3) are (5, 4).

Step 2: Calculate the slope of the two points.

Simply enter the two points into the slope formula to obtain the slope: (y2 — y1) / (x2 — x1). The slope of a line is calculated by dividing the distance of its vertical change by the distance of its horizontal change. Here’s how to calculate the slope of the line that connects the points (2, 5) and (8, 3)

(3–5)/8–2 = -2/6 = -1/3

The line has a slope of -1/3. To find this slope, reduce 2/6 to its simplest terms, 1/3 because both 2 and 6 are equally divisible by 2.

Step 3: Find the inverse of the slope of the two points.

To get the negative reciprocal of a slope, just take the slope’s reciprocal and flip the sign. Simply swapping the x and y coordinates and changing the sign yields the negative reciprocal of a number. 1/2’s reciprocal is -2/1, or simply -2; -4’s reciprocal is 1/4.

Because 3/1 is the reciprocal of 1/3 and the sign has been reversed from negative to positive, the negative reciprocal of -1/3 is 3.

Step 4: Write a line equation in slope-intercept form.

A slope-intercept equation is y = mx + b, where “x” and “y” represent any x and y coordinates in the line, “m” represents the slope of the line, and “b” represents the y-intercept of the line. The y-intercept is the point at which the line meets the y-axis. Once you’ve written this equation down, you may start looking for the equation of the perpendicular bisector of the two points.

Step 5: In the equation, enter the negative reciprocal of the original slope. The negative reciprocal of the slope of the points (2, 5) and (8, 3), respectively, was 3. Plug the 3 into the “m” in the equation y = mx + b.

y = mx + b = y = 3x + b = y = 3x + b

Step 6: You are already aware of the fact that the midpoint of the points (2, 5) and (8, 3) is (5, 4). Because the perpendicular bisector passes through the midpoint of the two lines, the coordinates of the midpoint may be plugged into the equation of the line. Simply enter (5, 4) into the line’s x and y coordinates.

(5), (4) → y = 3x + b = 3(5) + b = 15 + b

Step 7: Determine the intercept.

You have discovered three of the four variables in the line’s equation. You now have enough data to calculate the remaining variable, “b,” which is the y-intercept of this line. To determine the value of the variable “b,” just isolate it. Simply take 15 off both sides of the calculation.

4 = 15 + b

b = -11

Step 8: Write the perpendicular bisector equation.

To create the equation of the perpendicular bisector, just enter the slope of the line (3) and the y-intercept (-11) into the slope-intercept equation of a line. You should not enter any terms into the x and y coordinates since this equation allows you to locate any coordinate on the line by entering any x or y coordinate.

y = mx + b

y = 3x — 11

The perpendicular bisector of the points (2, 5) and (8, 3) has the equation y = 3x — 11.

Perpendicular Bisector Properties

A perpendicular bisector splits or bisects a line segment into two parts.
A perpendicular bisector intersects (or is perpendicular to) a line segment at right angles.
Each point on the perpendicular bisector is equidistant from both ends of a line segment.
When dealing with practical geometry, you will frequently encounter the use of perpendicular bisectors, such as when attempting to construct an isosceles triangle or determining the center of a circle.

Perpendicular Bisector Theorem

We apply the perpendicular bisector theorem when we have a perpendicular bisector in the picture.

The converse is also true: if a point is equidistant from the ends of a segment, it sits on the perpendicular bisector.

This theorem teaches us that there is a succession of locations equidistant from the ends of a line segment that, when joined together, may create a line to the middle of that line segment at a right angle.

How To Find Perpendicular Bisector?

It is fairly straightforward to find a perpendicular bisector. You may have guessed how to accomplish this from reading the part above that defines a perpendicular bisector.

An easy method to determine a perpendicular bisector is to measure the line segment you need to bisect. Then, divide the measured length by two to determine the midway. Draw a line at a 90-degree angle out from this midpoint. If you have a ruler, that’s all you need to do to find the perpendicular bisector!

Perpendicular Bisector: Definition, Theorem, Equation, And ​​How To Construct