NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.5 – Part 1
Perpendicular bisector
To find this point, you will construct three perpendicular bisectors, one for each side of the triangle. The point where all three perpendicular bisectors intersect is called the Circumcenter. Using this center point, we can draw a circle that passes through all three vertices.
1. ABC is a triangle. Locate a point the interior of which is equidistant from all the vertices of
Solution:

Draw perpendicular bisectors PQ and RS of sides AB and BC respectively of triangle ABC. Let PQ bisects AB at M and RS bisects BC at point N.

Let PQ and RS intersect at point O.Join OA, OB and OC.

Now in
[By construction]
[By construction]
[Common]
[By SAS congruency]
[By Coorresponding Parts of Congruent Triangles] …..equation(1)

Similarly,
[By Corresponding Parts of Congruent Triangles.] …..equation(2)

From eq. (1) and (2),

Hence O, the point of intersection of perpendicular bisectors of any two sides of equidistant from its vertices.