Here are two consequences of the parallel theory of B&B Chapter 4. (1) Two lines perpendicular to the same line are parallel (or the same line) and (2) if a transversal is perpendicular to one of two parallel lines it is perpendicular to both.
- Experiments
We begin with a visual phenomenon. Then we fill in with some relationships that go a long way to explain the phenomenon.
- Work through GTC Exploration 4.1.
- Work through the GTC Exploration 4.2. Then, working with a partner, answer on a separate sheet of paper the Questions. You will hand this in.
- Carry out an alternate version of 4.2. (1) Draw points O, X., Y, and A. Construct the lines OX and OY. Construct B as the reflection of A across line OX and construct C as the reflection of line OY. (2) Construct the segments AB, BC, CA and their midpoints M, Q and N. Also construct segments MN. NQ, QM and AQ. (3) Write down on your paper to hand in all the similarity and congruence relationships among triangle ABC and the four sub-triangles MBQ, MAQ, NCQ, NAQ.
- Explain why you get additional congruence relations when the lines OX and OY are perpendicular.
- Proofs
The goal of each of these proofs is the following.
Theorem R. Let ABC be a right triangle. If O is the midpoint of the hypotenuse BC, then AO = BO = CO.
In each case we will begin by constructing a right triangle ABC in this way. Construct line AB. Construct the line through A perpendicular to AB. Then let C be a point on this perpendicular line. Hide the lines and construct the segments that are sides of triangle ABC.
We will split up into pairs and do ONE or two of these proofs and share the results.
- Proof 1
. Start with the right triangle ABC. Construct the midpoints M, N and O of the sides AB, AC and BC. Construct segments OA, OM, ON. Now continue from this point to prove the relationships needed to prove the Theorem.
- Proof 2.
Start with the right triangle ABC. Construct O as the midpoint of BC and construct the perpendicular lines from O to AB and AC. Now continue from this point to prove the relationships needed to prove the Theorem.
- Proof 3.
Start with the right triangle ABC. Construct the perpendicular bisector m of AB and construct the point Q as the intersection of m and BC. Now continue from this point to prove the relationships needed to prove the Theorem.
- Proof 4.
Start with the right triangle ABC. Construct the perpendicular bisectors of AB and AC. Now continue from this point to prove the relationships needed to prove the Theorem.
- Carpenter's Theorem.
Let B and C be two points. The locus of points A so that angle BAC is a right angle is the circle with diameter BC (except for the points B and C themselves).
- Is this the same as Theorem R? How do the statements differ? Write down your answer.
- Carry out Exploration 4.3 and use this a basis for a proof of part of the Carpenter's theorem.
- Applications
- Construct the tangent to a circle through an exterior point, as explained on page 52. Look over the explanation on the preceding pages and carry out the Sketchpad experiment if you have time.