This lab consists of most of the Explorations from Chapter 4 of the Green Book (Geometry Through the Circle).
Work through Exploration 4.1. Specifically,
Investigation 1. Do this with a moving corner of notebook paper.
Investigation 2. Make the sketch and study the trace. Answer the questions as to what you conjecture the shape is, but NO EXPLANATION needed. This will come in the next section.
Investigation 1. Do this with a paper right triangle.
Investigation 2. Carry out this reflection construction and answer the questions.
Investigation. Carry out the investigation and answer the questions in the Conclusions. (You should be able to manage without the earlier Exploration 2.2 cited there.)
Construction Problems 1 and 2. Carry out both constructions. Notice how the same constructions can be done with a straightedge and compass.
Consider the following figure. Draw segment AB. Construct the line m through B which is perpendicular to line AB. Construct C as a point on this perpendicular line m. Hide m and construct segments BC and CA. This is a right triangle with right angle at B. Make a script from this figure.
Conjecture to be proved:
Let M be the midpoint of AC. Construct segment BB’. Based on observations from earlier explorations, what do you conjecture is the relationship among the lengths of segments MA, MB, and MC? You can construct do some measurements or construct the circle with center M through B and get some evidence for this conjecture, but the goal here is an explanation. The following figures will give some approaches for a proof.
In each case prove some relationships in this figure sufficient to prove the conjecture above. (You can use what you already proved in homework.) Notice that the points that are constructed initially in the figure differ each time, and that a point that appears to be a midpoint, for example, must be proved to be a midpoint if it is not constructed as a midpoint.
Figure 1. Construct the right triangle ABC as above (use the script). Let M, N, O be the midpoints of CA, AB, BC. Construct segments MN and MO and MB.
Figure 2. Construct the right triangle ABC as above (use the script). Let M be the midpoint of CA. Construct the lines through M perpendicular to segments BA and BC. Let A’ be the foot of the perpendicular to BA and let C be the foot of the perpendicular to BC.
Figure 3. Construct the right triangle ABC as above (use the script). Let P be the midpoint of BA. Construct the perpendicular bisector of side BA. Let the perpendicular bisector intersect AC at point Q.
This investigation can be carried out pretty well on paper, if time does not permit in the lab.
Investigation 1. Make the construction and study the angles. Answer Q1, Q2, Q3.
The main result is:
Carpenter Locus Theorem. Given 3 points A, B, C, angle ABC is a right angle if and only if B is on the circle with diameter AC.