If 8 points in a plane are chosen to lie on or inside a circle of diameter 2cm then show
that the distance between some two points will be less than 1cm.
how should i proceed for this?

Question

If 8 points in a plane are chosen to lie on or inside a circle of diameter 2cm then show
that the distance between some two points will be less than 1cm.
how should i proceed for this?

anonymous · Answer

*

shubhamsrg · Answer

will this help ?
suppose a pointP at random is chosen at a distance <1 from O
taking P as center and constructing a circle of radius 1, shaded reigon is the common region ,,as P gets closer to O,,space in which any other point can lie becomes lesser..
but how do i prove whole cirlce gets shaded after max 7 points ?

shubhamsrg · Answer

|dw:1344942451700:dw|

anonymous · Answer

hmm is there a definition for the points? what if all the eight points are on the circle and spaced closely.

shubhamsrg · Answer

well we have to prove there'll be min 2 of those 8 points which will have separation less than 1 cm..

anonymous · Answer

oh i read the question wrong, sorry. i will try it again.

anonymous · Answer

Area of the circle must be $\pi$. Each point could be imagined as a circle of area $\pi/4$, maybe?

shubhamsrg · Answer

i dont  get it?? come again..

anonymous · Answer

|dw:1344946376091:dw|
like you see there are maximum 7 points with distance >=1. If I put one more point it's distance with some point will be less than 1