Review Examination I

Geometric Structures, Mathematics 244

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

1. Draw a diagram of the following (when possible the diagram should be drawn in the plane):

a. A scalene triangle

b. An equilateral triangle

c. An equiangular 4-gon that is not a square

d. An equilateral quadrilateral that is not a square

e. A non-convex, non-self-intersecting 4-gon

f. A trapezoid with three equal sides that is not a rectangle

g. An equiangular hexagon which is not a regular hexagon

h. A self-intersecting regular pentagon

i. An isosceles right triangle

j. An acute angle triangle

k. A parallelogram which is not a rectangle or a rhombus

l. A kite

n. Draw a graph with 7 vertices and 9 edges

o. Draw an equiangular triangle

p. Draw an isosceles triangle which is not equilateral

q. A non-convex self-intersecting 7 sided polygon

r. Give some examples of 3-dimensional geometrical figures

s. Give some examples of 2-dimensional geometrical figures

t. What is the difference between a ray and a segment?

2. Give precise definitions of the following:

a. Convex set

b. A circle

c. A circular disc

3, a. From the point of view of analytical geometry, what are the points of the Euclidean plane?

b. From the point of view of analytical geometry, what are the lines of the Euclidean plane?

c. From the point of view of analytical geometry, what are the points of the taxicab plane?

4. a. Draw all the polyominoes which consist of 3 or fewer 1x1 cells.

b. Draw two pentominos of different perimeter.

c. Draw three hexaminos of different perimeter.

5. Draw a legal polyomino which is non-convex because it has a "hole."

6. Draw 5 trees which are "different" and which have exactly 8 vertices.

7. How many edges does each of the trees above have?

8. Draw three different nets for a cube.

9. For the graphs below give the number of vertices and edges, as well as the valence (degree) of each vertex. (The valence of a vertex is the number of edges of the graph at that vertex.)

10. Give examples of finite and infinite projective planes by describing the points and lines of such planes.

11. The points and lines below are in the real projective plane, though of as ordered triples of homogeneous coordinate real numbers.

Given:

Triangle: ABC with vertices A = (1, -5, 0), B = (2, 10, 2), C = (3,3,5)

and A'B'C' with vertices A' = (2, -22, 3), B'=(1, 14, 4), C' = (4, 4, -2)

a. Write down the Euclidean equivalent for each of these points if there is one.

b. Write down the equation of the line through A and A'

c. Write down the equation of the line through B and B'

c. Write down the equation of the line through C and C'

d. Is the hypothesis of Desargues' Theorem satisfied? If it is verify that the conclusion of the Theorem holds for these two triangles.

e. State Pappus' Theorem for the real projective plane.

12. Give an example of a finite hyperbolic plane. How many points and lines does the plane have?

13. a. Let Q be a quadrilateral with vertices A = (0, 0), B= (a, 0), C = (b, c), D = (d, e). What are the coordinates of the midpoints of the four sides? Find the slopes of the 6 lines determined by these 4 points. What do you notice?

b. Can you find choices of a, b, c, e, and e so Q is non-convex?

14. What are the three (incidence) axioms for:

a. An affine plane

b. A projective plane

c. A hyperbolic plane

15. a. Is it possible to define all the terms in a mathematical system?

b. What are the usual undefined terms for a geometry?

16. Draw a diagram of the Fano plane? How many points are there in this plane? How many lines? How many lines through each point?

17. Draw an affine plane with 4 points. How many lines does this plane have? How many lines are there through a point? If a parallel class in an affine plane consists of all the lines parallel to each other in the plane, list all the lines in each of the parallel classes of the plane that you draw.

18. Euclid's 5th postulate dealt with parallelism. Give a modern version of this axiom.

19. In Euclidean three space is it possible for lines in different planes not to have any point in common? (Lines of this kind are usually referred to as skewed lines.)

20. Draw three different nets for a cube which has no lid.