**Construction of circumcenter of a triangle :**

Even though students know what is circumcenter, many students do not know, how to construct circumcenter.

Here we are going to see construction of circumcenter.

**Key Concept - C****ircumcenter**

The point of concurrency of the perpendicular bisectors of the three sides of a triangle is called the circumcenter and is usually denoted by S.

To construct a circumcenter, we must need the following instruments.

1. Ruler

2. Compass

Let us see, how to construct circumcenter through the following example.

Construct the circumcenter of the triangle ABC with AB = 5 cm, <A = 70° and <B = 70°.

Step 1 :

Draw triangle ABC with the given measurements.

Step 2 :

Construct the perpendicular bisectors of any two sides (AC and BC) and let them meet at S which is the circumcentre.

From the steps of construction of circmcenter, first we have to know, how to construct perpendicular bisector.

So, let us see, how to construct perpendicular bisector.

The steps for the construction of a perpendicular bisector of a line segment are:

Step 1 :

Draw the line segment AB.

Step 2 :

With the two end points A and B of the line segment as centers and more than half the length of the line segment as radius draw arcs to intersect on both sides of the line segment at C and D.

Step 3 :

Join C and D to get the perpendicular bisector of the given line segment AB.

In the above figure, CD is the perpendicular bisector of the line segment AB.

This construction clearly shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This bisects the line segment (That is, dividing it into two equal parts) and also perpendicular to it. Since this bisects, it finds the midpoint of the given line segment.

The perpendicular bisector of a triangle is a line which is passing through the mid point of the side and also perpendicular to that side.

Step 1 :

Draw the triangle ABC.

Step 2 :

Select one the sides of the triangle, say AC

With the two end points A and C of the side AC as centers and more than half the length of the side AC as radius draw arcs to intersect on both sides of the side AC and join the points of intersection of the arcs.

Using the steps explained above, in the above triangle ABC, perpendicular bisector is drawn to the side AC.

Similarly we can draw perpendicular bisectors to the sides AB and BC as given below.

Hence, every triangle will have three perpendicular bisectors.

**Key Concept - C****ircumcenter**

The point of concurrency of the perpendicular bisectors of the three sides of a triangle is called the circumcenter and is usually denoted by S.

After having gone through the stuff given above, we hope that the students would have understood "Construction of circumcenter of a triangle"

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