Arithmetic Exercises

In the exercises below, assume that you are given pairs of points whose distance apart represents a length.

1. Addition and subtraction. Given segments of lengths $A$ and $B$, construct a segment of length $A+B$ or $A-B$.
2. Multiplication. Given segments of lengths $A$, $B$, and $1$, construct a segment of length $AB$.
3. Division. Given segments of lengths $A$ and $1$, construct a segment of length $1/A$. Using the result of the problem above, show that if you also have a segment of length $B$, you can construct a segment of length $B/A$.
4. Square roots. Given segments of lengths $A$ and $B$, construct a segment whose length is $\sqrt{AB}$. Clearly, if one of the segments has length $1$, you can construct a segment of length $\sqrt{A}$ using the same technique.
5. Square a rectangle. Given a rectangle, construct a square with exactly the same area.
6. Construct a pentagon. Construct a regular pentagon. For a hint, see the footnote².