# Problem Set 9 Straightedge and Compass Constructions

For each of the following straightedge and compass constructions, show all circles and lines used in the construction and highlight your final answer.
Given that 1, a, and b are constructible numbers whose lengths are given below, use straightedge and compass only to construct a triangle with side lengths 1, a, and b. Show all relevant circles. Highlight your final answer in color with thicker lines and deemphasize the circles and lines with thinner lines. Explain your strategy. Is your triangle unique?

Given that angles with measures and are constructible with straightedge and compass only, use straightedge and compass only to construct a triangle with angles of measures and . Show all circles and lines used in the construction explain your strategy. Is your triangle unique?

Two houses (located at points A and B) are set on one plot of land. The homeowners want to build a fence between the houses, but don't know where the property line is. Use straightedge and compass only to determine where the fence should go so that the fence is equidistant from each home.
How would the fence be altered if a third house (located at point C) is built on the plot of land?
The figure below shows a circle with center C and radius 1 and an angle of measure that was constructed with straightedge and compass only. Prove that a segment of length cos can be constructed with straightedge and compass only by constructing such a segment in the figure.
The figure below shows a circle with center C and radius 1 and a segment of length cos that was constructed with straightedge and compass only. Prove that an angle of measure can be constructed with straightedge and compass only by constructing such an angle in the figure. Write something short that explains your strategy and why it works.
The figure below shows a circle with center C and radius 1. Use straightedge and compass only to construct an equilateral triangle inscribed in the circle. Show all circles or arcs used in the construction and highlight your final answer. Write something short that explains your strategy and why it works.
The figure below shows a circle with center C and radius 1. Use straightedge and compass only to construct a square inscribed in the circle. Show all circles or arcs used in the construction and highlight your final answer. Write something short that explains your strategy and why it works.
The figure below shows a circle with center C and radius 1. Use straightedge and compass only to construct a regular hexagon inscribed in the circle.