Quiz 3 Answers

Math 445 Quiz 3 ANSWERS

Problem 1

In each of the figures 1A and 1B below, construct the perpendicular bisector of AB in the appropriate model. As usual, write a FEW main steps to make the construction clear. Do not write a lot.

FIGURE 1A

In this D-model figure, construct the D-perpendicular bisector of AB. (The point I is the one removed in the model.)

Construct the E-circle c through A, B, I. This is the D-line AB.
To construct the perpendicular bisector as an E-circle d, first construct the center P as the intersection of E-line AB (the center is on this line if A inverts to B) and the E-tangent line to c at I (d must be orthogonal to c at I).
Then d is the circle with center P through I.

FIGURE 1B

In this P-model figure, construct the P-perpendicular bisector of AB.

Label as p the boundary circle of the Poincare disk. Construct the supporting circle c of the P-line AB as the circle through A, B, and A', the inversion of A in p.
To construct the perpendicular bisector as an E-circle d, first construct the center P as the intersection of E-line AB (the center is on this line if A inverts to B) and the E line through the intersection points of c and p (d must be orthogonal to both p and c this line is the radical axis of c and p).
Then d is the circle with center P with radius = tangent segment from P to p (or tangent from P to c, same radius).

Problem 2

· Review of terminology: The circles of Apollonius of points A and B are the same as the hyperbolic pencil (hyperbolic coaxal family) with limit points A and B. These are the circles orthogonal to the elliptic pencil (elliptic coaxal family) of circles through A and B. The other kind of pencil or coaxal family is a parabolic pencil tangent to a given line at a given point A.

· The task: In each case in the table below, the family of objects is either a set of circles that form a pencil or else are arcs of such circles. For each case below, you are asked to tell what is the pencil? This means to tell (1) which of the 3 types of pencil and (2) tell specifically what are the special points A and B for the pencil). You may wish to sketch figures to help in your explanation.

· Notation: In these questions, the disk for the P-model is the interior of the circle c with center O and radius r. The DWEG model consists of points in the plane with the point I removed (and the point Infinity added).

Description of the family	What kind of pencil?	What are the 2 points A and B? (One point and line for parabolic)
(a) In the P-model, for a P-point G, what pencil is the set of P-lines through G?	Elliptic	G and G', inversion of G in P-boundary.
(b) In the D-model, for a D-point H, what pencil is the set of D-lines through H?	Elliptic	H and I. (I is the special point removed in the D-model.)
(c) In the P-model, for P-points U and V, what pencil is the set of P-circles with P-center U?	Hyperbolic	U and U'.
(d) In the D-model, for D-points X and Y, what pencil is the set of D-circles with D-center X?	Hyperbolic	X and I.
(e) In the P-model, for a P-line g, what pencil is the set of P-lines orthogonal to g?	Hyperbolic	Points at infinity on g, i.e., the intersection points of supporting circle of g and the boundary circle of the disk.
(f) In the D-model, for a D-line h, what pencil is the set of D-lines orthogonal to h?	Parabolic	Circles orthogonal to h at I.

Problem 3

Consider a figure with two lines AB and CD, with A and D on opposite sides of line BC.

Is this statement True or False?

Statement: For this figure in the hyperbolic plane, if angle ABC = angle DBA, then line AB is parallel to line CD.

· True or False? __TRUE______________

· If true, indicate why (the key idea without all the details). If false, indicate why or give a counterexample

If there were an intersection point X on ray BA, then by point symmetry, there would be another intersection point on CD so that triangle BXC is congruent to triangle CYB. But distinct lines cannot intersect in two points.

· State the converse of the statement. Is the converse true or false in the hyperbolic plane?

Converse: If line AB is parallel to line CD (with A and B on opposite sides of line BC), then angle ABC = angle DBA

Converse implies that there is only one parallel to line AB through C, so this is false in hyperbolic case.