Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto at Scarborough
MAT D02S
Solution #1
1. Parameterize by (t) = (2, t)
s=
2. .
Z
2
1/7
2 t 1/7.
2
1 1
0 + 12 dt = ln t1/7 = ln 2 ln(1/7) = ln 14
t
Given P , Q, to find the centre A:
AP Q is
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #3
1. (a) G is the centroid so A G = 21 AG = 12 2 = 1
(b) Set x = GD = B D. As in part a, GB = 2B G = 2(x + x) = 4x.
Power of the poi
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #4
1. Let A , B , C , D be the images of A, B, C, D under inversion in a circle of
radius r about O.
Since the circle shown containing A and B p
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Solution #2
1. AB = AC so ABC is isosceles and ABC = ACB.
BD, CE are angle bisectors, so DBC = 21 ACB.
Therefore DBC ECB (ASA), so BD = CE.
2. Since AB = 
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
DEPT. OF COMPUTER & MATHEMATICAL SCIENCES
UNIVERSITY OF TORONTO SCARBOROUGH
MAT D02S
Problem Set I
Due: Thurs., Jan. 23 at beginning of class
1. (page 133; #7.2)
2. We know that in the Poincare upper half plane model H, the lines are the vertical
Euclidea
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #5
1. Let F3 = # of triangular faces and let F4 = # of squarefaces
Then
2E = 4V
3F3 = V
4F4 = 3V
V E + F3 + F4 = 2
Therefore V 4V + 31 V + 43 V
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set III
Due: Thurs., March 4th at beginning of class
1. In Euclidean Geometry, consider the triangle ABC pictured below, and let
A and B be the midpoints of th
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set IV
Due: Thurs., March 21st at beginning of class
1. (page 144; #7.30)
2. Find a fractional linear transformation f (z) for which f (0) = 2, f (i) = 1 + i
a
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set V
Due: Thurs., April 4th at beginning of class
1. (page 164; #7.79) (originally Question 6 from Problem Set IV)
2. (page 110; #5.12)
3. (page 157; #7.65)
4