Perpendicular Bisector – Calculate it using slope and points

In mathematics, a term is used to divide the line into equal parts, that term is known as perpendicular bisector. In geometry, a perpendicular bisector is widely used for making a 90-degree line after intersecting it.

For dealing with the units, measurements, properties, lines, etc. geometry is used for such dealings in mathematics. The perpendicular bisector is that term used at a larger scale for making bisectors on triangles, lines, circles, etc.

What is a Perpendicular Bisector?

A segment or a line that is perpendicular to a line that goes through the norm of the segment is known as the perpendicular bisector. It is also stated that a term that distributes a line into congruent parts by giving equal ratio on both sides.

In other words, when a line or a segment cross from another line or a segment having equally distributed for two parts and makes a line of 90-degree, then this intersection of the lines or segments are called perpendicular bisector.

In the equation of the perpendicular bisector, two fixed points are used, two points and a slope of the line is involved and is written as,

y – y₁ = m (x – x₁)

The above equation of the perpendicular bisector has x and y as a fixed point, x₁ and y₁ as points, and m is the slope of the line by using the points.

For the calculation of the perpendicular bisector, we must be familiar with the slope of the line and points of the line. An online tool such as perpendicular bisector calculator can be used for the calculations of the perpendicular bisector and the slope by using points.

What is Slope?

The slope is a term used for the measurement of the steepness or sharpness of the given line along with the direction of the line. The slope is a term widely used for the calculation of the equation of the line by using various ways like slope-intercept form or the point-slope form.

Slope intercept form and the point-slope form are mostly used for the perfect calculation of the equation of the line, slope plays a very vital role in this. In perpendicular bisector, the slope performs the main role to calculate it. Without slope, we are unable to find the perpendicular bisector of the line.

Let’s understand how to calculate the slope with the help of examples, which is then used to find the perpendicular bisector of the line, triangle, or circle.

Example 1: For Positive points

Find the slope of the given points, (8, 4) and (9, 11)?

Solution

Step 1: Name the given points, in form of x and y.

x₁ = 8, x₂ = 9, y₁ = 4, y₂ = 11

Step 2: Now take the general equation of the slope.

Slope = m = (y₂ – y₁) / (x₂ – x₁)

Step 3: Put the values of each point in the slope equation to calculate it.

Slope = m = (11 – (4)) / (9 – 8)

Slope = m = (11 – 4) / (1)

Slope = m = 7/1

Slope = m = 7

Example 2: For positive and negative points

Find the slope of the given points, (-3, -15) and (23, 45)?

Solution

Step 1: Name the given points, in form of x and y.

x₁ = -3, x₂ = 23, y₁ = -15, y₂ = 45

Step 2: Now take the general equation of the slope.

Slope = m = (y₂ – y₁) / (x₂ – x₁)

Step 3: Put the values of each point in the slope equation to calculate it.

Slope = m = (45 – (-15)) / (23 – (-3))

Slope = m = (45 + 15) / (23 + 3)

Slope = m = 60/26

Slope = m = 30/13

Slope = m = 2.3077

How to Calculate Perpendicular Bisector by using points and slope?

Perpendicular bisector can be calculated easily by using the points and the slope. When two points are given such as (x₁, y₁) and (x₂, y₂) then we have to calculate the slope first by using these two points and taking the negative reciprocal of the calculated slope.

Then take the midpoints of the given points and then we take the general equation to calculate the perpendicular bisector by using one of two points and the calculated slope.

Example  

Find the perpendicular bisector of the given points, (8, 2) and (9, 11)?

Solution

Step 1: Name the given points, in form of x and y.

x₁ = 8, x₂ = 9, y₁ = 2, y₂ = 11

Step 2: First of all, we have to calculate the slope. So, take the general equation of the slope.

Slope = m = (y₂ – y₁) / (x₂ – x₁)

Step 3: Put the values of each point in the above slope equation to calculate it.

Slope = m = (11 – (2)) / (9 – 8)

Slope = m = (11 – 2) / (1)

Slope = m = 9/1

Slope = m = 9

Step 4: Find the negative reciprocal of the calculated slope.

Slope = m = -1/9

Slope = m = – 0.11

Step 5: Now take the general equation of the perpendicular bisector.

y – y₁ = m (x – x₁)

Step 6: Take the midpoints of the given points for using in the place of x₁ and y₁ of the equation.

x₁ = x₁ + x₂ / 2 = 9 + 8 / 2 = 17/2 = 8.5

y₁ = y₁ + y₂ / 2 = 11 + 2 / 2 = 13/2 = 6.5

Step 7: Place the values of midpoints along with the calculated slope in the above perpendicular bisector equation.

 y – 6.5 = -0.11(x – 8.5)

y – 6.5 = -0.11x + 0.9444

y = -0.11x + 0.9444 + 6.5

y = -0.11x + 7.4444

Summary

A perpendicular bisector is used in geometry for the conversion of the line into two equal parts. To calculate this the knowledge of finding the slope, negative reciprocal of the slope, and finding the midpoints are compulsory. Without these terms, we cannot find the perpendicular bisector.