Mt 322 Topics in Geometry - Fall 2003

Study Guide for Final Exam

Test date and time: Thursday, December 11 -- 10:30 am - 12:30 pm

The Final Exam will be a cumulative test over the work of the entire semester. As usual, this test will include some kind of construction using Sketchpad. Because of the schedule of exams during Exam Week, we must adhere to the two-hour timeframe for the exam. This test will be about the same length as the previous tests.

This test will cover material from Chapters 1 - 5, 7 - 8, and Appendix A. In particular, you should be able to:

• do some geometric constructions using Sketchpad,
• write a proof (or disproof) of a given statement (as appropriate),
• give a counter-example (disproof) of a statement if it is not true in general,
• set up a truth table and evaluate a Boolean expression,
• use basic area, perimeter, and volume formulas of geometry,
• work with linear expressions and equations,
• use algebra and analytic geometry as a tool in solving problems,
• use coordinates as a tool in proving or disproving a given geometric statement,
• work with functions that transform the plane -- that is, functions which take points of the plane as input, and return points of the plane as output,
• determine whether (or not) a particular transformation of the plane is an isometry -- and then prove that it is an isometry or explain why it is not an isometry,
• work with various one- and two-dimensional coordinate systems.

Basic Geometric Constructions

You should be able to perform the following constructions and explain (or prove) why they work:

• construct a perpendicular from point P to a line l,
• construct the perpendicular bisector of a given line segment,
• find the foot of the perpendicular from a point P to a line l,
• construct a tangent to a circle from a point P on the circle,
• construct a tangent to a circle from a point P exterior to the circle,
• construct the bisector of a given angle,

• define a rectangular, square, or polar coordinate system on a GSP worksheet,
• perform any of the above constructions on a coordinate grid as well as on a Sketchpad worksheet without the coordinate grid,
• define and plot a function in either Cartesian or polar form on a Sketchpad worksheet,
• modify the domain of the function (by selecting the function -- only the function -- then choosing Properties, then Plot in the Edit menu),

• perform constructions in the Poincare disk,
• construct a right triangle in the Poincare disk and measure the hyperbolic lengths of its sides,
• construct an asymptotic triangle in the Poincare disk (that is, a triangle with one, two, or all three vertices on -- or very nearly on -- the fundamental circle),
• construct a triangle in the Poincare disk and measure the size of the hyperbolic angles,
• construct a quadrilateral in the Poincare disk that has one, two, or three right angles,
• explain (or prove) why the angle sum of triangles in the Poincare disk is between 0 and 180 degrees,
• explain (or prove) why it is not possible to construct rectangles in the Poincare disk, and
• explain the importance of Euclid's fifth postulate for Euclidean geometry. (For example, what are some of the things that we take for granted in a Euclidean world that we seem to lose when we are working the the Poincare disk where Euclid's fifth postulate does not hold.)

Truth Tables and Boolean Expressions

The development of geometric proofs rests on the basic rules of logic. You should be able to

• set up a truth table for Boolean expressions involving AND, OR, and NOT,
• evaluate a Boolean expression involving AND, OR, and NOT, (that is, you should be able to determine whether the expression is true or false),
• set up a truth table for a Boolean expression involving an implication,
• identify the hypothesis and the conclusion of an implication statement (a proposition or a theorem),
• set up a truth table for an expression of the form P implies Q,
• negate a Boolean expression involving AND, OR, and NOT,
• negate a Boolean expression of the form P implies Q,
• negate a simple statement involving a universal or existential quantifier,
• use principles of logic or Boolean algebra to explain why a single counterexample is enough to disprove a universal statement.

Analytic Geometry

You should be able to:

• work with rectangular and polar coordinates in the plane,
• give several different polar coordinates for the same point in the plane,
• use basic unit circle trigonometry to solve problems involving points in the plane,
• use basic area, perimeter, and volume formulas of geometry,
• apply these basic formulas to solve perimeter, area, and volume problems for composite geometric figures constructed from simpler figures,
• use the Pythagorean theorem to find the lengths of the sides of a right triangle, and to calculate the distance between points in the plane,
• use various forms of the linear equation in solving problems involving points in the plane,
• express the slope of a line in various ways -- as "rise/run", as angle of inclination, and as the tangent of an appropriate angle,
• prove that the slope of a line is equal to the tangent of the angle of inclination,
• define and plot a function in either polar or rectangular coordinates on a Sketchpad worksheet, and
• use coordinate geometry to prove (or disprove) geometric statements about figures in the plane.

Isometries in the Plane

You should be able to:

• use the method of Activity 1, page 102, to construct a function that performs a transformation of the plane,
• determine whether (or not) a given transformation of the plane is one-one, onto, and/or distance-preserving,
• prove or disprove that a given function is (or is not) an isometry,
• identify whether a particular isometry (given in a geometric diagram or as an algebraic function) is a translation, a reflection, or a rotation,
• identify whether a given isometry has (or does not have) any fixed points, -- and find those fixed points if there are any,
• identify whether a given isometry is direct or opposite,
• identify or describe the inverse of a given isometry, and
• perform a composition of two isometries -- for example, a reflection followed by a rotation, and so on.

Symmetry in the Euclidean Plane

You should know and be able to:

• list the symmetries of a given figure in the plane, (such a figure could be a polygon, a letter of the alphabet, a geometric figure, a wallpaper design, or a frieze),
• find the complete set of symmetries of a particular polygon,
• use cycle notation to express each of the symmetries of a given polygon,
• calculate the composition of two symmetries,
• identify the identity symmetry in a group of symmetries,
• identify the inverse of each symmetry in a group of symmetries,
• give an example of a pair of symmetries which are commutative -- that is, give an example from the group of symmetries of the square or triangle which are commutative,
• give an example of a pair of symmetries from the group of symmetries of the square or the triangle which are not commutative -- that is, give a counter-example to show that, in general, composition of symmetries is not commutative, and
• prove or disprove that a given subset of symmetries of a figure is a group.

Basic Algebra

For this exam, the material in Chapter 4 and Appendix A requires that you can do basic algebraic, arithmetic, and trigonometric computations. Converting from rectangular coordinates to polar coordinates (and vice versa) requires being able to work with the unit circle and the two special triangles. (See the Benchmark #2 Study Guide for a more explicit list of basic algebra skills.)

Developing a Proof

The test will include several opportunities to prove or disprove a particular geometric construction or statement.

• You should be able to prove that your strategy for doing a particular geometric construction is correct.
• You should be able to use the Pythagorean theorem in a proof involving lengths and right angles.
• You should be able to use the idea that the angle in a semi-circle is a right angle in a proof.
• You should be able to identify and use similar triangles in a proof.
• You should be able to identify whether two given triangles are (or are not) congruent. You should know (and be able to apply) the criteria that are acceptable for proving that two given triangles are congruent and/or similar in the Euclidean plane.
• You should be able to prove whether or not a particular transformation, given as a function f: (x, y) --> (u, v), is an isometry. That is, you should be able to prove/disprove that a given function is one-one, onto, and distance-preserving.