WHAT YOU MAY ASSUME (MED 383w, Fall 06)

You may use the facts, formulas, and theorems below in your solutions unless you are being asked to derive the formula or a special case, explain why the theorem or a special case is true, etc. If you think there is something I have left off this list that is needed to solve a problem, please ask me.

Geometry

Lines and angles

• There are 180° in a straight angle.
• When two lines intersect, the vertical angles formed are congruent.
• If two parallel lines are cut by a transversal, then the alternate interior angles are congruent and the corresponding angles are congruent.
• If a transversal crosses two lines so as to make the corresponding angles (or alternate interior angles) congruent, then the two lines are parallel.
• The perpendicular bisector of a line segment is the set of all points equidistant from the endpoints of the segment.

Triangles

• The sum of the measures of the angles of a triangle is 180°.
• The Pythagorean Theorem
• The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
• The side-side-side, side-angle-side, and angle-side-angle conditions for congruence of two triangles.
• Two angles of a triangle are congruent if and only if the sides opposite those angles are congruent. (Isosceles triangle conditions)
• Two triangles are similar if and only if their corresponding angles are equal.
• Two triangles are similar if and only if their corresponding sides are all in the same ratio.
• If two triangles have two angles of one congruent, respectively, to two angles of the other, the triangles are similar.
• If two triangles have an angle of one congruent to an angle of the other and the included sides are proportional, then the triangles are similar.
• The altitude to the hypotenuse of a right triangle forms two right triangles similar to each other and to the original triangle.
• If a line is drawn between two sides of a triangle and parallel to the third side, then the new triangle formed is similar to the original triangle.
• The area of a triangle is half the product of the height and the width.

Circles

• Two central angles in a circle with equal intercepted arcs have equal measure.
• The ratio of the measure of a central angle of a circle to 360° is equal to the ratio of the length of the intercepted arc to the circumference. (Use 2π instead of 360° if you are using radian measure)
• An angle inscribed in a semicircle is a right angle.
• The ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degrees in the angle of the sector to 360°.
• (Use 2π instead of 360° if you are using radian measure)
• The measure of an angle inscribed in a circle is half the
• measure of the intercepted arc, when the arc is measured in radian or degree measure.
• An angle formed by a tangent and a chord to a circle is measured by half its intercepted arc.
• A tangent to a circle is perpendicular to the radius to the point of tangency.
• The perpendicular bisector of a chord of a circle
• lies along a diameter of the circle.
• Formulas for the area and circumference of a circle.

Parallelograms

• Opposite sides of a parallelogram are congruent.
• The diagonals of a parallelogram bisect each other.
• The diagonals of a rhombus are perpendicular.

Trigonometry

• The definitions of the trig functions, both in terms of right triangles and in terms of coordinates of points on circles.
• The relationships between the trig functions -- e.g., tan a = (sin a)/(cos a).
• The trig identities coming from the Pythagorean Theorem: sin²x + cos²x = 1, etc.
• The law of sines.
• The law of cosines.
• Addition and double angle formulas

Analytic Geometry

• The definition of slope in terms of rise and run (change in y over change in x).
• What it means for (a,b) to be on the graph of an equation (i.e., the equation is true when a and b are substituted for x and y, respectively) or on the graph of a function.
• A parabola with vertical axis and is the graph of a function of the form
f(x) = ax² + bx + c.
• How translating a graph changes the equation for the graph.
• Basics of polar coordinates: what they are, and how to go back and forth between polar and rectangular coordinates.

Algebra

• How to expand a product of expressions and collect terms.
• The quadratic formula.
• How to solve equations and inequalities in standard/reasonable cases. (This is necessarily vague; if in doubt, ask.)

Arithmetic Concepts

• Place value concepts (e.g., the digit in the hundreds column stands for that many hundreds)
• Concepts of "divides evenly", "is a factor of", "is a multiple of", "leaves a remainder of"

Calculus

• Standard formulas for derivatives and integrals
• Differentiation formulas for sum, product, quotient
• Chain rule
• Derivative tests for finding maxima and mimima
• Definition of derivative and integral
• The area under a curve is represented by a certain integral
• Use of derivatives for speed, acceleration.
• Use of derivatives for testing for increasing, decreasing, points of inflection, concavity