please help with this math problem i need it. help?

Question

four points are chosen in a circle of a radius 1 unit.show that at least two of the points are within square root 3 of one another.

Answer 1

The farthest apart the 4 points can be from each other is when they are forming a square. (That is the central angle subtended by the two closest points is 90degrees.) (The example given above is the N,S,E & W location.) Otherwise if you move one point further from its one neighbor, then it will be closer to the other neighbor. The closest distance will even be smaller than the previous one.

In such a case, the straight distance between them is √(1²+1²) = √2 < √3.


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Answer 2

Let's call the four points A, B, C, and D.
Pick an arbitrary point on the circle. Call this point A.
Next, draw a circle of radius sqrt(3), centered on A.

If it's possible for four points in the unit circle to all be more than sqrt(3) from each other, then points BCD must all lie within the crescent-shaped area of the unit circle that lies outside the sqrt(3)-radius circle centered on A.

Now, place point B at one of the tips of the crescent.
Draw a circle of radius sqrt(3), centered on B.

If it's possible for four points in the unit circle to all be more than sqrt(3) from each other, then points C and D must lie within the unit circle, but outside of the two sqrt(3)-radius circles centered on A and B.

However, all points within this remaining area are less than sqrt(3) distance from each other. Since this area must contain both points C and D, it must be impossible to have four points in the unit circle all be more than sqrt(3) from each other.

Hence, if you pick four points in a unit circle, at least two of the points must be within sqrt(3) of each other.

Hopefully that helps ya!

Answer 3

the farthest apart they could be is at the top, bottom, left, and right of the circle. the circumference is 2pi, so they would be 2pi/4 = pi/2 apart and pi/2 < sqrt(3). note that the curved length of the perimeter will always be longer than the straight length of a line segment between the two points.