Question

two equal circles each of radius r, intersect each other such that each circle passes through the centre of the other. The length of the common chord of the circles is what ?

Answer 1

let the circles intersect at 2 point say A,B and centres of circles be C and D
the common chord is AB and it is perpendicular bisector of CD let intersection point of AB AND CD be P
now,CP=PD=r/2 and APC=90
and AC =r as A is a point on circle
as APC =90,IN TRIANGLE APC
by pythagoras theroem
Ap^2+PC^2=AC^2
r^2=(r/2)^2+X^2
X^2=3r^2/4

$x = \frac{ \sqrt{3} }{4} r$