| course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
27 |
|
Paper 2, Section II, F |
2016 |
(a) Let $A B C$ be a hyperbolic triangle, with the angle at $A$ at least $\pi / 2$. Show that the side $B C$ has maximal length amongst the three sides of $A B C$.
[You may use the hyperbolic cosine formula without proof. This states that if $a, b$ and $c$ are the lengths of $B C, A C$, and $A B$ respectively, and $\alpha, \beta$ and $\gamma$ are the angles of the triangle at $A, B$ and $C$ respectively, then
$$\cosh a=\cosh b \cosh c-\sinh b \sinh c \cos \alpha .]$$
(b) Given points $z_{1}, z_{2}$ in the hyperbolic plane, let $w$ be any point on the hyperbolic line segment joining $z_{1}$ to $z_{2}$, and let $w^{\prime}$ be any point not on the hyperbolic line passing through $z_{1}, z_{2}, w$. Show that
$$\rho\left(w^{\prime}, w\right) \leqslant \max \left{\rho\left(w^{\prime}, z_{1}\right), \rho\left(w^{\prime}, z_{2}\right)\right}$$
where $\rho$ denotes hyperbolic distance.
(c) The diameter of a hyperbolic triangle $\Delta$ is defined to be
$$\sup {\rho(P, Q) \mid P, Q \in \Delta}$$
Show that the diameter of $\Delta$ is equal to the length of its longest side.