Math 445 Final Exam

Problem 1: Matrix of a transformation

Let ABC and ACD be two equilateral triangles in the (x, y) plane, with the common side AC. If A = (0,0) and B = (1,0), find the matrices for the following transformations T and M.

(a) T = the rotation with center A that rotates ABC to ACD.

(b) M = the line reflection that reflects each triangle to the other

(c) Compute the matrix of MT and tell what transformation this matrix defines.

Problem 2: Coordinates and ratios

In the figure, let Q be the intersection of lines BU and CV, Let W be the intersection of line AQ with line BC. Let Z be the intersection of line UV with line BC.

1. Find the barycentric coordinates of Q?

2. Tell what equality of ratios is needed to determine whether points W and Z divide BC harmonically. Then verify whether or not WZ does divide BC harmonically.

3. Find the barycentric coordinates of W?

4. Find the barycentric coordinates of Z?

Problem 3: Proofs concerning inversion

A. Given a Euclidean circle c with center O and points P and Q distinct from O, let P' and Q' be the inversions of P and Q in c. Prove that triangle OPQ is similar to OQ'P'.

B. Given a Euclidean circle c and a Euclidean line m, let m' be the inversion of m in c. If m does not pass through the center of c, what kind of object is m'? [You may include the point at infinity as a point of m if this simplifies your answers.] Prove your statement.

C. Given the same circle and points P and Q as in (a), is it true or not that that image of triangle OPQ, including its sides, is triangle OQ'P", including its sides? Give a clear explanation of your answer.

Problem 4: Ellipse

(a) Given two points A and B and a distance d > |AB|, the set of points P for which |AP|+|PB|=d is an ellipse. In the figure below, construct a "random" point P on the ellipse and also the tangent line p to the ellipse at P.

(b) Explain why the construction you used constructs a point P of the ellipse.

(c) Explain why the construction you used constructs a tangent line p of the ellipse.

Problem 5: P-model

(a) In The P-model figure below (the center of the P-model is O), given two P-points A and B, construct the P-circle with P-center A through B.

(b) Construct a P-point C and P-lines AB, BC, CA so that P-triangle ABC is equilateral.

Section B. Short Answer

1. If ABC is a triangle in hyperbolic non-Euclidean Geometry and X, Y, Z are the midpoints of the sides. Which triangle has bigger angular defect, ABC or XYZ? ____

2. True or false. In hyperbolic non-Euclidean Geometry geometry, the perpendicular bisectors of the sides of a triangle are always concurrent. ____

3. True or false. Given any two disjoint Euclidean circles d1 and d2, there is a circle c such that the image of d1 and d2 by inversion in c consists of a circle and a line. ____

4. True or false. In hyperbolic non-Euclidean Geometry, an equilateral triangle is equiangular. ____

5. What do the triangles in Desargues theorem look like if two pairs of corresponding sides intersect at infinity? (Sketch or paint a word picture.) ____

6. True or false. In affine geometry, any circle can be mapped by an affine transformation to a hyperbola. ____

7. For this circle and point P, construct the polar of P using only a straightedge (no compass).