Overview of Section 1.4 More on Constructions
Constructions: Constructions possible with a compass and straightedge.
Three classic problems of antiquity;
Trisecting an angle
Doubling the cube
Squaring the circle
1. These constructions are impossible with compass and straightedge.
2. This does NOT mean that an angle equal to one-third of any given angle does not exist. It only means the angle can not be constructed with compass and straightedge.
3. The proofs of non-constructiblity may require some advanced mathematics; it is an interesting history of mathematics topic.
This section is about problems that may be constructible.
Investigation and analysis. "Take the Problem as solved" stragegy for analysis. Way of discovering a path to a proof.
Construction. Perform the construction with a compass and straightedge.
Proof. Reasoning that the construction follows from what we know about our axiomatic geometry system.
Construction 1.7: Equal Distances
Comment: View the line m as one of the two lines given in Problem 22, Sect 1.3 and k as the constructed angle bisector. Why not vice versa?
Why are X and Y on opposite sides of Point A?
Are there any other points equidistant from A and line m?
Construction 1.8: Extending a line beyond an obstruction.
Given a line AB on one side of an obstruction. Extend AB across the obstruction without touching the obstruction.
GSP File for Construction 1.8.
Construction 1.9: A line through a point in the interior of an angle
Clarification. We are given an angle with vertex at A and a point P on the interior of the angle. The task is to constuct a segment XYwith X on side of the angle and Y on the other so that P is the midpoint of XY. That is, find X and Y such that PX = XY.
Open GSP File
Construction 1.10: An inaccessible vertex of an angle.
Construct a line through point P on the interior of an angle to an inaccessible vertex of the angle.
Open GSP file for Exploration
Is this construction similar to the one done in Problem 22, Section 1.2?
Open GSP file for one completed construction
NOW SOLVE THIS 1.15
Three explorations of Problem 22, Section 1.2
Construction 1.11: SSA construct a triangle given two sides and an angle opposite one of the sides.
In our notation the construction is
b, c, angle at B
Open GSP file
NOW SOLVE THIS 1.16
This problem essentially investigates Construction 1.11. Use the GSP file and drag the ends of the given line segments to explore the questions.
Problem Set 1.4