] X,Y and Z are three points on a unit sphere (i.e. a sphere with radius 1). The space distance between any pair of the three points is √ 2 (i.e. XY = YZ = ZX = √ 2). An ant can only crawl on the surface of the sphere. The ant starts from point X, passes through points Y and Z, and then returns to X. What is the shortest possible distance of this trip?

help would be appreciated!

Guest Jun 14, 2021

#1**0 **

Say A and B are two of the points on the sphere, and that O is the center of the sphere and OH is the perpendicular bisector of AB. Then the image you see above shows the intersection of the sphere and a plane through the three points O, A, and B. The ant can minimize the distance between A and B by walking on the arc AB. So if we had the length of this arc, we could multiply it by 3 to find the total minimum distance a smart ant would choose to traverse. So help the ant and find the needed arc length (Hint: what is the measure of the angle AOB, given that \(AH=\frac{\sqrt{2}}{2}\), and that OA is 1 unit long?

Guest Jun 14, 2021