Question

prove  that  point(7,5) is equidistant  from points  (2,4) and(6,10)

Answer 1

let points be A(7,5)  B(2,4)  C(6,10)
      THE DISTANCE BETWEEN A AND B=√(7-2)²+(5-4)²
                                                            =√25+1=√26 UNITS
    THE  DISTANCE BETWEEN A AND C=√(7-6)²+(5-10)²
                                                           =√1+25=√26 UNITS.
     since AB=AC..A IS EQUIDISATANT FROM B AND C

Answer 2

A(2,4),  B(6,10)              P(7,5)

let us do without calculating squares and square roots.

So P lies on the perpendicular bisector of AB, if  PA = PB.
Mid point C of AB is =  C(4, 7).     
Slope of AB = 6/4 = 3/2
Slope of PC = 2/-3  = -2/3    Product of slopes is -1.
So CP is perpendicular bisector of AB.  It P is equi distant from A and B