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.
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.
- 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.
- 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.
- Opposite sides of a parallelogram are congruent.
- The diagonals of a parallelogram bisect each other.
- The diagonals of a rhombus are perpendicular.
- 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
- 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.
- 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.)
- 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"
- 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