Perpendicular Bisector

A perpendicular bisector is a line that intersects a line segment at its midpoint, divides it into two equal parts, and forms a right angle (90°) with it at the point of intersection.

In the figure shown below, the perpendicular bisector divides the line segment AB into two halves at its midpoint.

Properties

• A perpendicular bisector divides a line segment into two equal parts at its midpoint.
• It intersects the line segment at a right angle (90°).
• Every point on the perpendicular bisector is equidistant from the endpoints of the segment.
• It passes precisely through the midpoint of the line segment.
• For a given line segment, there exists a unique perpendicular bisector.

Construction of a Perpendicular Bisector

A perpendicular bisector of a line segment can be constructed using a ruler and a compass. It divides the given line segment into two equal parts at its midpoint and forms a right angle (90°) with it.

Steps to Construct a Perpendicular Bisector

• Draw a line segment XY of any suitable length.
• With X as the centre and a radius more than half of XY, draw arcs above and below the line segment.
• With Y as the centre and the same radius, draw arcs to intersect the previous arcs.
• Mark the points of intersection as P and Q.
• Join P and Q. This line PQ is the perpendicular bisector of XY. It intersects XY at point O, which is the midpoint.

Equation for a Perpendicular Bisector

To find the equation of a perpendicular bisector of a line segment:

• Find the midpoint of the given line segment.
• Determine the slope of the line segment.
• Find the negative reciprocal of this slope (this is the slope of the perpendicular bisector).
• Use the point-slope form: y - y1 = m(x - x1) to write the equation using the midpoint.

Learn More: Equation of a Straight Line

Perpendicular Bisector in a Triangle

A perpendicular bisector of a triangle is a line that divides a side into two equal parts at its midpoint and is perpendicular (90°) to it. Each triangle has three such bisectors, one for each side. These bisectors intersect at a single point called the circumcenter, which is equidistant from all three vertices and acts as the centre of the circumcircle.

Steps of Construction

• Draw a triangle and label the vertices as A, B, and C.
• With B as the centre and a radius greater than half of BC, draw arcs above and below the side BC.
• With C as the centre and the same radius, draw arcs intersecting the previous arcs.
• Mark the points of intersection as X and Y, and join them to form the perpendicular bisector of BC.
• Repeat the same process for sides AB and AC.
• The three perpendicular bisectors intersect at a point called the circumcenter.

Perpendicular Bisector Theorem

The perpendicular bisector theorem states that any point lying on the perpendicular bisector of a line segment is equidistant from its endpoints.

In the above figure, points Q, R, S, and T lie on the perpendicular bisector of line segment MN. Therefore:

• MQ = NQ
• MR = NR
• MS = NS
• MT = NT

This shows that every point on the perpendicular bisector is at an equal distance from both endpoints.

Related Articles

• Perpendicular Lines
• Angle Bisector
• Parallel Lines
• Lines Parallel to the Same Line

Solved Examples

Example 1. Draw a 6 cm line and construct a perpendicular bisector on it.

Solution:

Below is the line of 6 cm with perpendicular bisector:

👁 Screenshot-2023-11-14-at-60349-PM

Example 2. Draw a 10 cm line and construct a perpendicular bisector on it.

Solution:

Below is the line of 10 cm with perpendicular bisector:

👁 Perpendicular Bisector Example 2

Example 3. Draw an equilateral triangle and draw a perpendicular bisector for the sides of the triangle.

Answer:

Below is the equilateral triangle with perpendicular bisector:

👁 Perpendicular Bisector Solved Example 3

Example 4. Draw a line segment of AB of 7cm and construct a perpendicular bisector on it.

Solution:

Below is the line segment of 7 cm with perpendicular bisector:

👁 Screenshot-2023-11-14-at-60046-PM

Example 5. Draw a line segment and construct a perpendicular bisector on it.

Solution:

Below is the line segment with perpendicular bisector:

👁 Screenshot-2023-11-14-at-55027-PM

Example 6: Find the equation of the perpendicular bisector of a line segment with endpoints at (-3, 1) and (5, 7).

Solution:

Step 1: Calculate the midpoint:

Midpoint = ((-3 + 5) / 2, (1 + 7) / 2) = (1, 4).

Step 2: Determine the slope of the line segment:

Slope (m) = (7 - 1) / (5 - (-3)) = 6/8 = 3/4.

Step 3: Find the negative reciprocal of the slope for the perpendicular bisector:

Perpendicular Bisector Slope = -4/3

Step 4: Use the midpoint (1, 4) and the calculated slope to write the equation of the perpendicular bisector:

y - (4) = -4/3x + 4/3

y = −4/3​x + 4/3 ​ + 4

y = −4/3​x + 16​/3

The equation of the perpendicular bisector is y = −4/3​x + 16​/3

Practice Problems

Q1. Draw a 5 cm line and construct a perpendicular bisector on it.

Q2. Draw a 9 cm line and construct a perpendicular bisector on it.

Q3. Draw an equilateral triangle and draw a perpendicular bisector for the sides of the triangle.

Q4. Draw a line segment of 10 cm and construct a perpendicular bisector on it.

Q5. Draw a line segment of 6 cm and construct a perpendicular bisector on it.