hyperbolic geometry

Question

1. Consider in hyperbolic geometry:  Suppose that lines 𝓁 and m intersect at O. Suppose also that P and Q are points on m such that O*P*Q. Let P ' and Q ' be the feet of the perpendiculars from P and Q to 𝓁 respectively. Prove that PP' < QQ'.

Answer 1

For this problem, we will use the properties of perpendicular lines and hyperbolic lines. We are given two lines l and m. They intersect at the point O. So, the point O is on both l and m. The two points P and Q are on the line m. The problem also tells us that O*P*Q. That is the point P is between O and Q. The points P' and Q' are on l and they are the feet of the perpendiculars from P and Q respectively.

Now use the hyperbolic property: The perpendicular distance from a point to a line is the shortest considering all the lines from the point considered to points on the line and the it is unique.. That means, if I draw a perpendicular line from the point P to the line l then that line, PP'is the shortest of all lines that I can draw from P to the line l. It is same with the point Q. However, since the point Q is further along the line m from O than P, the perpendicular from Q to l at Q' is longer than the perpendicular from P to l at P'. So,we conclude that PP' is shorter than QQ'. That is, PP' < QQ'.