NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.5 – Part 3
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Intersection of perpendicular bisectors
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.
Q3 In a huge park, people are concentrated at three points (see figure):

Where there are different slides and swings for children.

Near which a manmade lack is situated.

Which is near to a large parking and exit.
Where an ice  cream parlor should be set up so that maximum number of persons can approach it?
Solution:

The parlor should be equidistant from A,B and C.

For this let we draw perpendicular bisector say l of line joining points B and C also draw perpendicular bisector say m of line joining points A and C.

Let l and m intersect each other at point O.

Now point O is equidistant from points A, B and C.

Join OA, OB and OC.

Proof:
In and ,
[Common]
[P is the midpoint of BC]
[By SideAngleSide congruency]
[By Corresponding Parts Congruent Triangles] …..equation (1)

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

From eq. (1) and (2),

Therefore, ice cream parlour should be set up at point O, the point of intersection of perpendicular bisectors of any two sides out of three formed by joining these points.