## shubhamsrg 4 years ago 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?

1. anonymous

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2. shubhamsrg

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 ?

3. shubhamsrg

|dw:1344942451700:dw|

4. anonymous

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

5. shubhamsrg

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

6. anonymous

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

7. anonymous

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

8. shubhamsrg

i dont get it?? come again..

9. anonymous

|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

10. anonymous

@Ishaan94 @shubhamsrg

11. anonymous

got it?

12. anonymous

|dw:1344946685043:dw|

13. shubhamsrg

cool.. maybe this is satisfactory enough!! thanks..

14. anonymous

yw