Perpendicular Bisector

Key Concepts

• Perpendicular Bisector Theorem.
• Concurrency.
• Circumcenter.

Introduction 

In the previous session, we have learned about Midsegment Theorem and Coordinate Proof. 

In the previous classes, we have learned about segment bisectors and perpendicular lines. 

Now we will learn Perpendicular bisector Theorem, Concurrency, and Circumcentre. 

Do you hear about Perpendicular bisector? 

Do you have any idea about segments? 

Midsegment Theorem 

A segment, ray or line or a plane that is perpendicular (90 degrees) to a segment at its midpoint is called Perpendicular Bisector. 

A point on a perpendicular bisector is equidistant from the two end points of the segment. 

Perpendicular Bisector theorem states that in a plane, if a point is on perpendicular bisector, then it is equidistant from two end points of the segment. 

Perpendicular Bisector Theorem

In a plane, if a point is on perpendicular bisector, then it is equidistant from the end points of the segment. 

If CP is perpendicular bisector of AB, then CA = CB 

Converse of Perpendicular Bisector Theorem 

In a plane, if a point is equidistant from the end points of a segment then the point lies on the perpendicular bisector. 

If DA = DB, then D lies on the perpendicular bisector of AB. 

Example 1: 

If XV is the perpendicular bisector, UV = 8x + 14, VW = 5x + 35. Find WV. 

Solution: 

Given XV is perpendicular bisector, 

We use perpendicular bisector theorem, 

UV = VW 

8x + 14 = 5x + 35 

8x – 5x = 35 – 14 

3x = 21 

x = 21/3  

x = 7 

Now, WV = 5x + 35  

 = 5*7 + 35  

= 35 + 35  

= 70 units. 

Example 2: 

In the given figure if XV is perpendicular bisector of UW and YU = 40, YW = 40  

Find whether Y is on the perpendicular bisector or not?  

Solution: 

Given UY = 40, YW = 40 

Both the lengths are same 

Now we use the converse of perpendicular bisector theorem, 

If the lengths are equidistant, then the point is on perpendicular bisector. 

Therefore, Y is on the perpendicular bisector XV. 

Activity 

We are drawing perpendicular bisectors for each side of a triangle,  

Will they meet at least once, if so, where do they meet? 

Let us do an activity to know this. 

Activity

Take a paper and cut it in the shape of triangle,  

Draw the perpendicular bisectors for the triangles. 

Now fold the triangle cut, at which all the perpendicular bisectors meet. 

Mark it as P if ABC is the triangle 

Then measure AP, BP, CP lengths 

What will you observe from these measures? 

The point at which AP, BP, CP meet is called point of concurrency. 

The lengths AP, BP, CP are equal.  

Concurrency 

If three or more rays, lines, or segments intersect at the same point, then they are known as concurrent lines, rays, or segments.  

The point of intersection of the lines, rays, or segments is called the point of concurrency. 

Concurrency of Perpendicular Bisector of a Triangle 

Concurrency Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.  

If DP, EP, FP are perpendicular bisectors then AP=BP=CP 

Example: 

A point P is equidistant from all the vertices of a triangle A, B, C. Find the location of P such that it is equidistant from vertices. 

Solution: 

Given A, B, C are vertices of a triangle. 

P is a point equidistant from A, B, C. 

Now we need to find the location of P 

We use concurrency of perpendicular bisector theorem, 

From the theorem, we can say that, to find the point of location we need to use perpendicular bisectors. 

First, we draw the triangle ABC. 

Now we construct the perpendicular bisector for each side by using ruler and protractor. 

Then the point of concurrency is the location of P. 

Circumcenter 

The circumcenter of a triangle is defined as the point of concurrency of the three perpendicular bisectors of a triangle. 

The circumcenter is equidistant from all the three vertices,  

The center of a circle that passes through all three vertices is also known as circumcenter. 

From the figure, we can say that the location of circumcenter depends on the type of triangle.  

The circle with the circumcenter is said to be circumscribed about the triangle. 

Real Life Example 

Three students sat in the exam at three different points in the exam hall, the teacher is equidistant from these three students. Find the location of teacher.  

Solution: 

Consider three students as three vertices of a triangle. 

Teacher is equidistant from three students. 

Now we need to find the location of Teacher. 

We use concurrency of perpendicular bisector theorem, 

From the theorem, we can say that, to find the point of location we need to use perpendicular bisectors. 

First, we draw the triangle connecting three students. 

Now we construct the perpendicular bisector for each side by using ruler and protractor. 

Then the point of concurrency is the location of Teacher. 

 Exercise

• Find AC

• Point P is inside ABC and is equidistant from points A and B. On which segment must P be located?

• Find PC.

• Find AP.

• If P is equidistant from all the vertices of triangle ABC. A circle drawn with P as center touching all the vertices, then the center of the circle is also called ______
• In a scalene triangle, the circumcenter lies ____
• In an obtuse angled triangle, the circumcenter lies ___
• Prove the converse of perpendicular bisector theorem.
• Can you do an activity to show the location of circumcenter in a acute angled triangle?
• Identify the theorem used to solve the problem?

Concept Map

What have we learned

• Perpendicular bisector Theorem.
• Find segment lengths.
• Concurrency.
• Circumcentre
  

Frequently Asked Questions (FAQ’s): 

1. At which point the perpendicular bisectors of a triangle meet?  
   Circumcentre of the triangle is the point in which the perpendicular bisectors of the triangles meet, and it is equidistant from the vertex. 

2. What is the Perpendicular Bisector Theorem?  
   If a point is on the perpendicular bisector of a segment, then it is equidistant from the segments endpoints is the perpendicular bisector theorem. 

3. What is the difference between bisector and perpendicular bisector? 
   A line when bisects the line segment AB and is also perpendicular to it is known as perpendicular bisector whereas, bisector only bisects the line segment. 

4. Does the perpendicular bisector pass through the midpoint? 
   Yes, a perpendicular bisector passes through the midpoint as it cuts the line segment into two equal halves. 

5. What is the difference between bisection and intersection? 
   Intersection means dividing the line into certain ratio and bisection means dividing the line into equal ratio.