Perpendicular Bisector

Perpendicular Bisectors are the lines that bisect the other line at right angles. Perpendicular bisectors are a fundamental building block in Euclidean geometry. The perpendicular bisector divides a line into two halves and is equidistant from the endpoints. The concept of a perpendicular bisector is fundamental for constructing various geometric shapes and solving problems involving lines and segments. Perpendicular bisectors are often represented as a dashed line with a right angle symbol to indicate their perpendicular nature.

In this article, we will discuss what is perpendicular bisector, the equation for a perpendicular bisector, the construction of a perpendicular bisector, and the perpendicular bisector theorem in detail.

What is a Perpendicular Bisector?

Perpendicular bisector is a geometric construction that guarantees a line, ray, or line segment intersects another line segment at a right angle while also dividing it into two halves. It plays a vital role in geometry, trigonometry, and engineering serving as a basis for various geometric proofs and calculations. A perpendicular bisector divides another line segment into two equal parts while forming a right angle (90 degrees) with it. We can consider a perpendicular bisector to cut the original line in half and always perpendicular to the original line.

Perpendicular Bisector Definition

Perpendicular Bisector is a line or line segment that intersects another line or segment at a right angle splitting it into two equal parts. It’s a geometric concept used in mathematics and engineering to find the midpoint of a line segment and create right angles.

Perpendicular Bisector Properties

Some of the properties related to Perpendicualr Bisector are:

• Midpoint: The perpendicular bisector of a line segment passes through the midpoint, effectively dividing the line segment into two equal parts.
• Right Angles: This geometric element forms right angles with the line segment it bisects, as a consequence of the definition of perpendicularity.
• Equidistant: Points along the perpendicular bisector are equidistant from the endpoints of the line segment. This distance is consistently equal to half the length of the line segment.
• Uniqueness: There exists only one perpendicular bisector for a given line segment. The bisector’s unique determination is attributed to the midpoint of the line segment.
• Bisects Chords in a Circle: In the context of circles, the perpendicular bisector of a chord (a line segment connecting two points on the circle) passes through the center of the circle.
• Application in Geometry Construction: Perpendicular bisectors play a crucial role in geometry constructions, aiding in tasks such as finding the midpoint of a line segment or creating right angles.

Equation for a Perpendicular Bisector

The equation for a perpendicular bisector involves several steps.

• First, find the midpoint of the line segment AB by averaging the x-coordinates and y-coordinates of points A and B.
• Then, calculate the slope of line AB.
• To find the slope of the perpendicular bisector, take the negative reciprocal of the slope of AB.
• Using the midpoint and the new slope, you can write the equation of the perpendicular bisector in the point-slope form, which is y – y₁ = m(x – x₁).
• The equation for a perpendicular bisector of a line segment AB can be found by first determining the midpoint of AB which is taken as ‘m’.
• Then, calculate the negative reciprocal of the slope of line AB.
• Finally, you can use the point-slope form to write the equation where (x, y) is a point on the perpendicular bisector and m is the negative reciprocal of the slope of AB.

Learn More, Equation of a Straight Line

How to Draw a Perpendicular Bisector?

Below are the steps to drawing a perpendicular bisector:

Step 1: Start by finding the midpoint of the given line segment.

Step 2: Then, use a straightedge or a ruler to draw a line that passes through this midpoint and is perpendicular to the original line segment.

Step 3: This newly drawn line is the perpendicular bisector.

Step 4: To draw a perpendicular bisector for a line segment AB, locate the midpoint ‘m’ of AB.

Step 5: Then, use a protractor or compass to create a 90-degree angle with AB at point M. This will be the perpendicular bisector.

Perpendicular Bisector Theorem

Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a line segment then it is equidistant from the endpoints of the segment. In other words, the distance from the point to each endpoint is the same.

The Perpendicular Bisector Theorem is an important geometric principle that mentions that any point lying on the perpendicular bisector of a line segment is equidistant from the two endpoints of the segment. This theorem is a efficient for solving geometric problems and proving congruence in triangles.

Construction of a Perpendicular Bisector

Construction of perpendicular bisector is simple. To construct a perpendicular bisector, you can locate the midpoint of the line segment. Then, use a compass to draw two equal circles with their centers at the endpoints of the segment. Where the circles intersect is the intersection point of the perpendicular bisector.

Below are the systematic steps that can be followed to construct a perpendicular bisector:

• Start by locating the midpoint of the line segment using the average of the coordinates of the endpoints.
• Then, use a protractor or a compass to create a 90-degree angle at the midpoint.
• The line extending from this angle is the perpendicular bisector of the given segment.

Perpendicular Bisector Line Segment

‘Perpendicular bisector line segment’ refers to a line segment that is drawn perpendicular to another line segment and intersects it at the midpoint. The perpendicular bisector divides the original segment into two congruent parts ensuring equal distances to the segment’s endpoints.

This concept is essential in many mathematical and engineering applications. A perpendicular bisector divides a line segment into two equal halves while forming a right angle with the line segment.

Perpendicular Bisector of line

When you refer to the perpendicular bisector of a line, it usually means the line that is perpendicular to the original line and intersects it at the midpoint. This creates two equal line segments on either side. A perpendicular bisector is drawn so that it is both perpendicular to the given line and passes through its midpoint.

It effectively divides the original segment into two equal parts. Perpendicular bisector of line is a fundamental tool in geometry and is used in various mathematical and engineering applications.

How do You Find the Perpendicular Bisector?

To find the perpendicular bisector of a line segment, calculate the midpoint of the segment, and then determine the negative reciprocal of the original segment’s slope. With the midpoint and the slope, you can write the equation of the perpendicular bisector. To calculate a perpendicular bisector, you can follow below steps:

• Find the midpoint of the line segment using the average of the coordinates of the endpoints.
• Determine the slope of the original line segment.
• The negative reciprocal of the slope gives the slope of the perpendicular bisector.
• Use the midpoint and the calculated slope to write the equation of the perpendicular bisector line.
• Another method is to use the distance formula to find the midpoint and then calculate the perpendicular bisector’s equation.
• The perpendicular bisector can also be calculated using vectors by finding the direction vector of the line.

Perpendicular Bisector in a Triangle

In geometry, the perpendicular bisector of a side in a triangle is a line that divides that side into two equal segments and is perpendicular to it. When the perpendicular bisectors of all three sides of a triangle are extended, they intersect at a single point known as the circumcenter. The circumcenter is a significant point in a triangle because it is equidistant from all three vertices of the triangle.

This property of the circumcenter depict that it is the center of the unique circle that can be drawn to circumscribe or pass through all three vertices of the triangle. This circle is called the circumcircle. The circumcircle with the circumcenter as its center, touches each vertex of the triangle making it important in geometry and trigonometry.

Solved Examples on Perpendicular Bisector

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:

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:

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

Answer:

Below is the equilateral triangle with perpendicular bisector:

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

Solution:

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

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

Solution:

Below is the line segment with perpendicular bisector:

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, -3).

Step 2: Determine the slope of the line segment:

Slope (m) = (-7 – 1) / (5 – (-3)) = -1.

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

Perpendicular Bisector Slope = 1.

Step 4: Use the midpoint (1, -3) and the calculated slope to write the equation of the perpendicular bisector:

y – (-3) = 1(x – 1).

y + 3 = x – 1.

y = x – 4.

The equation of the perpendicular bisector is y = x – 4.

Example 7: Determine the length of the perpendicular bisector of a line segment with endpoints at (1, 1) and (9, 7).

Solution:

Step 1: Calculate the midpoint:

Midpoint = ((1 + 9) / 2, (1 + 7) / 2) = (5, 4).

Step 2: Determine the distance between the endpoints:

Distance = √((9 – 1)² + (7 – 1)²) = √64 + 36 = √100 = 10.

The length of the perpendicular bisector is half of this distance:

Length = (1/2) × 10 = 5 units.

So, the length of the perpendicular bisector is 5 units.

Practice Problems on Perpendicular Bisector

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 perpendicular bisector for the sides of triangle.

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

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

Perpendicular Bisector – FAQs

1. What is a Perpendicular Bisector?

A perpendicular bisector is a line, ray or line segment that intersects another line/line segment at a right angle while simultaneously dividing it into two equal parts.

2. What is the relationship between a Perpendicular Bisector and the Line?

A perpendicular bisector forms a right angle with the line or segment it bisects.

3. What is the Circumcenter of a Triangle?

The circumcenter is the point where the perpendicular bisectors of a triangle’s sides intersect.

4. Can you have a Perpendicular Bisector in Non-Euclidean Geometries?

Perpendicular bisectors are primarily defined in Euclidean geometry but similar concepts exist in other geometries.

5. How is the Perpendicular Bisector used in Architectural Design?

Perpendicular Bisector helps ensure symmetrical layouts evenly spaced elements and right angles in architectural plans.

6. In how many parts does a Perpendicular Bisector divide a line?

A perpendicular bisector divide a line/line segment/ray into two parts.

7. How can you find the Perpendicular Bisector of a Line Segment?

To find the perpendicular bisector of a line segment, first determine the midpoint of the segment. Then, draw a line perpendicular to the segment at the midpoint. This new line is the perpendicular bisector.

8. What is the relationship between Perpendicular Bisector and the Circumcenter of a Triangle?

The perpendicular bisectors of the three sides of a triangle intersect at a point known as the circumcenter. This point is equidistant from the triangle’s vertices making it an important element in triangle geometry.