Midterm Exam

This exam is open notes, so please use them. Make sure you cite which theorems and definitions you use. Don't use other people. You will have one week to complete the exam. There are 50 points possible on the exam.

1. (6 points) Let \(l\) and \(m\) be two lines. Prove that \(l \cap m\) has either 0,1, or \(\infty\) many points.
2. Let \(A\) and \(B\) be distinct points. Prove:
   1. (3 points) \(\overline{AB} = \overline{BA}\);
   2. (3 points) \(\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{AB}\);
   3. (3 points) \(\overrightarrow{AB} \cap \overrightarrow{BA} = \overline{AB}\).
3. Prove that the following are convex:
   1. (3 points) \(\{A\}\);
   2. (3 points) \(\overline{AB}\);
   3. (3 points) \(\overleftrightarrow{AB}\).
4. Let \(S\) and \(T\) be two convex sets. Prove or disprove:
   1. (3 points) \(S \cup T\) is convex;
   2. (3 points) \(S \cap T\) is convex.
5. (4 points) Prove that the hypotenuse of a right triangle is always the longest side.
6. (6 points) Prove that the shortest distance from a point \(A\) to a line \(l\) is along the line perpendicular to \(l\) passing through \(A\).
7. Define the distance between two points \((x_0,y_0)\) and \((x_1,y_1)\) in \(\mathbb{R}^2\) by \[D((x_0,y_0), (x_1,y_1)) = \max\{|x_1-x_0|, |y_1-y_0|\}.\]
   1. (6 points) Show that \(D\) is a metric.
   2. (4 points) Sketch \(\{(x,y) \in \mathbb{R}^2 \,|\, D((0,0),(x,y)) = 1\}\).