Comments and some Answers for Midterm

1. Proof: A Locus (25 points)

The locus is the circle with diameter AB, except for points A and B.

This is a theorem that one set equals another, so that means that it is an if and only if theorem that requires the proof one a statement and also the converse statement.

• This is Locus Theorem 6 of B&B and a proof can be found there.
• The proof that APB is a right angle implies that P is on the circle with diameter AB is problem 3.9 of Assignment 3B.
• In the course, this theorem was explored in Lab 3 , and Chapter 4 of GTC.

2. Problem: A Distance (15 points)

In this case, triangle ACB is similar to triangle to B'CA' but NOT similar to triangle A'CB'. The proof is by SAS by checking the equal ratios: CB'/CA = CA'/CB. This ratio = 3/ab = k the scaling factor, so also k = A'B'/7. This shows A'B' = 21/ab.

Note: The letters a and b are numbers, so they are given as any possible numbers and stay the same throughout the problem. You cannot change them or solve for them.

3. Construction: Tangents (15 points)

• This problem is noted as important in Assignment 3B, at the end.
• The construction was given in Lab 3, and can be found in GTC, Chapter 4, page 52. This construction was made a tool during the lab.
• The construction is also found in the construction chapter of B&B.
• The construction was reviewed and used in Lab 5 in constructing common tangents to two given circles.

4. Construction: Circles (20 points)

There are two circles to be constructed. The centers of the circles are the intersections of the line through B perpendicular to line m with the two lines that bisect the angles formed by m and n (i.e., the locus of points equidistant from m and n). One the centers are constructed, since the circles are drawn to pass through B.

This problem is related to

• Locus Theorem 5 of B&B, which is the same as Problem 2.6 of Assignment 2B and also related to problem 2.5. It is also related to problem 2.1 of Assignment 2A.
• Inscribed and tangent circles were also studied in Lab 2 and reviewed in Lab 5 construction practice.

5. Problem: Ratios in a trapezoid (25 points)

This problem is a combination of thales figures (dilations) with transversals. The answers are (a) 4/13 (dilation, then transversal), (b) 4/13 (dilation) (c) 4/17 (transversals and addition of segments) (d) 1 (problem 3.3 of Assignment 3A).

There are many examples of such ratio arguments in B&B, pp. 64-65 and the problems about constructing ratios such as problem 3.7 (Angle Bisector Ratio Theorem) of Assignment 3B.

Also see Lab 5 (Dilations) and Trapezoid ratio interactive page linke to Week 4.