A perpendicular bisector is widely used in geometry. Geometry is a branch of mathematics that deals with relationships, measurements, and properties of surfaces, points, angles, and lines. Perpendicular bisector divides the line into two equal parts. We can find the perpendicular bisector of lines, circles, triangles, etc. To find the accurate result of any point we can use an online Perpendicular Bisector Calculator.

What is a Perpendicular Bisector?

A perpendicular bisector is defined as the division of a line or a line segment into two equal parts or two congruent parts. The term bisect is used to express division equally. A perpendicular bisector is a line, segment, or ray which intersects another ray, line, or segment into two congruent parts at ninety degrees (90 degrees).

The equation of perpendicular bisector is,

y – y1 = m (x – x1)

We can find the perpendicular bisector of line, triangle, and circle. Let us discuss them briefly.

Perpendicular Bisector of a line segment

The perpendicular bisector of a line segment is formed if a line is perpendicular to another line and cuts it into two congruent parts. Let us take a figure to understand this concept.

By using a ruler and a compass we can easily calculate the perpendicular bisector on a line segment. The constructed perpendicular bisector cuts the given line segment into two equal parts exactly at its midpoint and makes two congruent line segments.

To construct a perpendicular bisector of a line segment we have to follow some steps. Let us discuss them briefly.

  • Draw a line segment AB of any appropriate length.
  • Take a compass, and with A as the center or middle with more than half of the line segment AB as width, draw arcs at the upper and lower side of the line.
  • Redo the same step with B as the mid or center.
  • Label the points of cuts or intersection as C and D.
  • Combine the points C and D. The point at which the perpendicular bisector cuts the line segment AB is its middle point or midpoint.

Practically,

The perpendicular bisector of a Circle

The perpendicular bisector of a circle is formed if a line is perpendicular to the chord of a circle and passes through the midpoint of the circle and cuts it into two congruent parts. Let us take a figure to understand this concept.

By using a ruler and a compass we can easily calculate the perpendicular bisector on a circle. The constructed perpendicular bisector cuts the given circle into two equal parts exactly at its midpoint.

To construct a perpendicular bisector of a circle we have to follow some steps. Let us discuss them briefly.

  • Draw a circle of any radius.
  • Draw a line segment AB of any diameter of the circle.
  • Take a compass, and with A as the center or middle with more than half of the line segment AB as width, draw arcs at the upper and lower side of the line.
  • Redo the same step with B as the mid or center.
  • Label the points of cuts or intersection as C and D.
  • Combine the points C and D. The point at which the perpendicular bisector cuts the circle AB is its middle point or midpoint.

Perpendicular Bisector of a Triangle

The perpendicular bisector of a triangle is formed if a line is perpendicular to the side of the triangle and crosses it through the midpoint. And cuts it into two parts. Let us take a figure to understand this concept.

Perpendicular bisector should pass through the midpoint of the sides. A perpendicular bisector doesn’t need to pass through the vertex of a triangle. Perpendicular bisector can be drawn on all sides of the triangle and where all the three bisectors meet is called the circumcenter of the triangle.

To construct a perpendicular bisector of a triangle we have to follow some steps. Let us discuss them briefly.

  • Draw a triangle and label each vertex a name, as X, Y, AND Z.
  • With A as a center and more than half AC as radius, draw arcs upper and lower of a line segment, AC. Repeat the same process without any change in campus with C as the center.
  • Label the points of cuts (intersection) of arcs as D and E respectively and combine them. This is the perpendicular bisector of one side of the triangle AC.
  • Repeat the same process for sides BC and AB. All the three perpendicular bisectors make an angle of 90 degrees at the midpoint of each side.

Properties of Perpendicular Bisector

Perpendicular bisectors can bisect a line or the sides of a triangle or a line segment. The properties of a perpendicular bisector are mentioned below.

  • Perpendicular bisector makes an angle of 90 degrees with the line that is being bisected.
  • Perpendicular bisector divides the sides of the triangle into equal parts.
  • Perpendicular bisector divides a line or a line segment into two equal parts.
  • Perpendicular bisector intersects the segment exactly at its midpoint.
  • The point of meeting of the perpendicular bisectors in a triangle is called its circumcenter.

How to Calculate Perpendicular bisector?

Let us take an example to understand the concept hoe to calculate the perpendicular bisector.

Example

Evaluate the points A (3,6) and B (8,8) to find the perpendicular bisector at x=2.

Solution

Step 1:First of all, find the midpoints of the line.

Midpoint of a line = x1+x2/2, y1+y2/2

Midpoint of AB = 3+8/2, 6+8/2 = (11/2, 14/2)

Step 2:Now find the slope of the line AB.

Slope (m) = y2-y1/x2-x1

Slope of AB = 8 – 6/8 – 3 = 2/5 = 0.4

Step 3:Find the slope of the line AB of perpendicular bisector (XY).

Slope of perpendicular bisector = -1/slope of the line

Slope of perpendicular bisector = -1/0.4 = -2.5

Step 4:apply formula to find the equation of perpendicular bisector.

y-y1 = m(x-x1)

y-14/2 = -2.5(x- 11/2)

Simplify, we get

y – 7 = -2.5(x- 11/2)

y = -2.5x + 20.75

This is the equation of perpendicular bisector

Step 5:Put x=2.

y = -2.5(2) + 20.75

y = 15.75

Summary

Now you are witnessed that this topic is not difficult. Once you grab the knowledge of this topic you can easily solve any problem related to the Perpendicular bisector.