Mt 322 Topics in Geometry - Fall 2002
Department of Mathematics & Computer Science
CARDINAL STRITCH UNIVERSITY
Sr. Barbara E. Reynolds, Ph.D.
Study Guide for Test #2
Test date: Tuesday, November 26
This test will cover major concepts in Chapters 4, 5, and 7. In particular,
you should be able to:
- do some geometric constructions,
- 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
- 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
- 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
- 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, and
- modify the domain of the function (by selecting the function -- only
the function, then choosing Properties, then Plot in
the Edit menu).
- Analytic Geometry
You should be able to:
- work with rectangular and polar coordinates in the plane,
- use basic unit circle trigonometry to solve problems involving points
in the plane,
- use the Pythagorean theorem 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, in terms of the tangent of an appropriate angle,
- prove that the slope of a line is equal to the tangent of the angle of
- 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 85, 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 compostion of two isometries -- for example, a reflection
followed by a rotation, and so on.
- Symmetry in the 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 identy 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, and
- 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.
- Basic Algebra
For this test, the material in Chapter 4 does require that you can do
basic algebraic and arithmetic 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 Study Guide
for a more explicit list of basic algebra skills.)
- Developing a Proof
The test will include three 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
- 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.
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This page was updated on November 19, 2002.