Planet Cutting

Geometry Level 2

On a spherical, land-only planet, there are two cities A and B on the equator.

A railway track--connecting these two cities and going around the planet--can possibly split the planet into two hemispheres, as shown in the diagram.

Given any two cities on this planet, is it always possible to construct an around-the-planet railway track that splits the planet into two hemispheres of equal size?

Yes, always Only if the two cities are on the equator Only if the two cities share the same lattitude Only if the two cities share the same longitude

Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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2 solutions

Linkin Duck
Jul 16, 2017

Three points: 2 random points on the surface and the point at the centre of the globe always define a plane that cuts through the centre, hence divides the giant sphere into 2 identical parts.

1 Helpful 0 Interesting 1 Brilliant 0 Confused
Richard Costen
Jul 14, 2017

Here is my attempt at a solution: To cut a sphere in half, you draw a circle around its surface which has the largest circumference, which happens when the circle's radius is the same as the sphere's radius. So just draw a circle through A and B using the centre of the sphere as its centre. Always possible since the 3 points are coplanar.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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