COURSE OUTLINE
MATH 10E
2014-2015
I.
GEOMETRY
 Circles
 Definitions: radius, congruent circles, chord, diameter, central angle, major and minor
arc, semicircle, secant, tangent, concentric, internally and externally tangent circles,
common internal and external tangents
 Proofs and applications of the following theorems:
 If a tangent intersects a line through the center of a circle at the point of tangency,
then they are perpendicular.
 If a line is perpendicular to a line through the center of a circle at a point on the
circle, then it is a tangent.
 Tangent segments that share an endpoint in the exterior of a circle are congruent.
 In the same or congruent circles, congruent chords have congruent major and minor
arcs.
 In the same or congruent circles, congruent arcs have congruent chords.
 If a line through the center of a circle is perpendicular to a chord, then it bisects the
chord and its arcs.
 In the same or congruent circles, congruent chords are equidistant from the center.
 In the same or congruent circles, chords that are equidistant from the center are
congruent.
 Definitions:
 The (degree) measure of a minor arc is the measure of its central angle.
 The (degree) measure of a major arc is 360 – the measure of its minor arc.
 The (degree) measure of a semicircle is 180.
 Intercepted arc.
 Proofs and applications of the following theorems:
 The measure of an inscribed angle is one-half the measure of its intercepted arc.
 Inscribed angles that intercept the same or congruent arcs are congruent.
 A quadrilateral is cyclic if and only if its opposite angles are supplementary.
 An angle inscribed in a semicircle is a right angle.
 The measure of a tangent-chord angle is half the measure of its intercepted arc.
 The measure of a chord-chord angle is half the sum of the measures of its
intercepted arc and the intercepted arc of its vertical angle.
 The measure of a secant-secant angle is half the difference of the measures of its
intercepted arcs.
 The measure of a secant-tangent angle is half the difference of the measures of its
intercepted arcs.
 The measure of a tangent-tangent angle is half the difference of the measures of its
intercepted arcs.
 If two chords intersect inside a circle, then the products of the lengths of their
segments are equal.
 If two secant segments share an endpoint in the exterior of a circle, then the
products of the lengths of the secant segments and their exterior segments are equal.
 If a tangent segment and a secant segment share an endpoint in the exterior of a
circle, then the products of the lengths of the secant segment and its exterior
segments equals the square of the length of the tangent segment.
 Area related to circles
 Review:
 The ratio of the areas of two circles is the square of the ratio of their radii.
 Area of sectors and segments of a circle.
 Constructions
 Prove informally how to construct:
 A segment congruent to a given segment.
 An angle congruent to a given angle.
 The bisector of an angle.
 The perpendicular bisector of a segment.
 A line perpendicular to a given line through a given point on the given line.
 A line perpendicular to a given line through a given point not on the given line.
 A line parallel to a given line through a given point not on the given line.
 The incircle of a given triangle.
 The circumcircle of a given triangle.
 Points that divide a given segment into three or more congruent segments.
 A tangent to a given circle through a given point on the circle.
 A tangent to a given circle through a given point not on the circle.
 Definitions: concurrent, centroid, orthocenter
 Prove and apply the following theorems:
 The medians of a triangle trisect each other.
 The medians of a triangle are concurrent.
 The altitudes of a triangle are concurrent.
 Prove informally how to construct:
 The centroid of a given triangle.
 The orthocenter of a given triangle.
 Coordinate Geometry
 Using coordinate geometry to prove statements and solve problems that are not
initially in a coordinate geometry setting
 Find the area of a polygon whose vertices’ coordinates are given.
II. TRANSFORMATIONS
 Review Functions
 Definition of function and relation.
 Examples of functions including linear, quadratic.
 Domain and range; interval notation.
 Inverse functions.
 Function composition.
 Absolute value functions
 Cubic functions, y = x3
 Piecewise functions, including floor   x   , ceiling   x  
 Line reflections and line symmetry; line reflections in the coordinate plane. Graphs of
inverse functions are reflections in the line with equation y  x .
 Rotations, including point reflections; rotations on the coordinate plane; rotational and
point symmetry.
 Translations, including on the coordinate plane. Can a non-linear shape be its own image
under a translation?
 Composition of transformations.
 Even and odd functions.
 Dilations.
III. EXPONENTS AND LOGARITHMS
 Properties of exponents.
 Review of exponential functions.
 Logarithmic functions as inverses of exponential functions.
 Properties of logarithms.
 Applications:
 Logarithms as a computational tool.
 Exponential growth and decay.
 Using logarithms in graphing.
 The number e.
 Solving logarithmic and exponential equations, including equations with extraneous
solutions.
IV. TRIGONOMETRY
 Review of trigonometry of the right triangle.
 Trigonometric functions as circular functions.
 Radian measure.
 Graphing trigonometric functions: periodic functions; continuous functions; amplitude;
frequency; period.
 Transforming trigonometric functions; i.e., comparing the graph of y  a  b sin  cx  d 
to the graph of y  sin x . Include that the graph of y  cos  x  2  is identical to the








graph of y  sin x .
Properties of trigonometric functions: comparing trigonometric functions of  to
trigonometric functions of  , 180   , 180   .
Inverse trigonometric functions.
Pythagorean, quotient, and reciprocal identities.
Co-function identities.
Formulas for trigonometric functions of a sum and difference, double-angle, half-angle
(include tan 2  1sincos and tan 2  1sincos ).
Proving trigonometric identities.
Solving trigonometric equations: linear and quadratic, also equations using trig
identities.
Law of Cosines; Extended Law of Sines: sina A  sinb B  sincC  2 R , where R is the
circumradius, abc  4KR ; solving triangles.
 Optional, Law of Tangents:
a b
a b

tan 12  A B 
tan 12  A B ) 
 Trigonometry applications to physics.
 Solving triangles, including the ambiguous case.
 Use of the graphing calculator
V. PROBABILITY
 Review of sample space, outcome set, events, counting principle, probability involving
“and” and “or”, Inclusion-Exclusion Principle, permutations, combinations, applications
involving permutations and combinations.
 Bernoulli experiments.
 Binomial Expansion Theorem – combinatorial proof.
 Conditional probability, including Bayes’ Theorem, and Law of Total Probability.