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- Geometry: the line segment between point A and point B
- Algebra: length of the segment
- Geometry: if and only if they do not intersect
- Algebra: having the same slope
- 1. To draw a straight line from any point to any point
- 2. To produce a finite straight line continuously in a straight line
- 3. That all right angels equal one another
- 4. That all right angles equal one another
- 5. That, if a straight line falling on two straight lines makes the interior angel on the same side less than the two right angels, the two straight lines, if produce indefinitely, meet on that side on which are the angles less than the two right angles
Implications of the Parallel Postulate
- "If m is a line and P is a point not on m,then there is exactly one line through P that is parallel to m"
- "rectangles exist"
Alternate Interior Angle Theorem
"If parallel lines are cut by a transversal, then alternate interior angles are congruent."
Variations of Parallel Postulate
- #1: Given a line m and a point P not on m, then there is no line through P that is parallel to m
- #2: Given a line and a point P not on m, then there are at least two lines through P that are parallel to m
The study of geometry where one is not allowed to use the Parallel Postulate nor any one of its variations
- if they have the same length = (~ on top)
- two angles are congruent if they have the same measure
- two polygons are congruent if you can match up their vertices in sequence so that all corresponding sides and all corresponding angles are congruent
if you can match up their verticies in sequence so that all corresponding angles are congruent, and so that all corresponding sides are in the same proportion
anytime the triangles angles are congruent, the triangles are similar.
Isosceles Triangle Theorem
if two sides of triangle are congruent, then the angles opposite those sides are congruent.
Exterior Angle Theorem
The exterior angle theorem says that the measure of an exterior angle of a triangle is equal to the sum of the measures of the other two remote interior angles.
The medians of a triangle are concurrent at a point called the centroid which also lifes two-thirds of the distance along each median from its vertex end. The angle bisector of a triangle are concurrent at the incenter. The perpendicular bisectors of the sides are concurrent at the circumcenter.
Law of Sines
- Triangle ABC
- sin (A) / a = sin (B) / b = sin (C) / c
Law of Cosines
- Triangle ABC
- c^2= a^2 + b^2 - 2ab(cosC).
Converse of Pythagorean Theorem
a^2 + b^2 = c^2 in triangle ABC then triangle ABC has a right angle at C
the shortest distance between two points is a straight line
Area of Triangle
- A = 1/2bh
- b is the length of the base and h is the length of an altitude drawn to that base.
Trignometric Formula: A = 1/2absinC
- Heron Forumla: based on the side lengths and the semiperimeter: s = 1/2(a+b+c):
- A = square root of s(s-a)(s-b)(s-c)
classical constructions are those that can be performed with a compass. (copying arcs of a given fixed length), and an unmarked straightedge (for drawing or extending lines if you are given two points). These were the geometry tools used by the ancient greeks.
What does it mean for a line to lie in a plane?
a line lies in a plane if every point on the line is also a point in the plane
What does it mean for a line to be perpendicular to a plane?
a line l is perpendicular to a plane P if l intersects P at one point A and if l is perpendicular to any line through point A that lies in plane P.
What does it mean for a line to be parallel to a plane?
a line is parallel to a plane if the line and the plane do not intersect.
What does it mean for two planes to be parallel?
Two planes are parallel if they do not intersect. Also, two planes are parallel if they are both perpendicular to the same line.
What does it mean for two planes to be perpendicular?
two planes are perpendicular if they intersect at a right angle. Supose plane P is perpendicular to line l and plane Q is perpendicular to line m. then plane P is parallel to plane Q if lines l and m are parallel, and P is perpendicular to q if l is perpendicular to m.
Explain why skew lines cannot intersect?
skew lines cannot intersect because they lie on parallel planes, which by definition do not intersect.
In what ways can two or three lines intersect on a plane?
- On a plane, two lines could be parallel or they could intersect in one point.
- Three lines could all be parallel, or two could be parallel or two could be parallel and one a transversal or any two of them could intersect in distinct points or the three lines might be concurrent at a single point.
In what ways can two or three lines intersect in space?
in space, two lines could be parallel or sew in which they do not intersect. if they are niether parallel nor skew, then the two lines intersect in one point.
Three lines could exhibit the same behavior as in the previous problem, or if two of th elines are skew then the third line could intersect one of them, both of them or neither of them.
In what ways can two or three plane intersect in space?
In space, two planes could be parallel or else they could intersect in a line. Three planes could all be parallel or two could be parallel and one could intersect each of teh other in a line or all three planes could intersect in a single point (like two walls and the floor intersect at the corner of a room) or the three planes could intersect in a line or the three planes could intersect two at a time which the three resulting lines of intersection being parallel.
In what ways can a line a plane intersect in space?
Either a line is parallel to a plane or else it intersects that plane in a single point.
How many points determine a line? a plane? space?
Two points determine a line. Three non-collinear points determine a plane. Four non-coplanar points determine space.
What are the volume and surface area of a cube?
If a cube has side length s, then it's volume is s^3 and its surface area is 6s^2 because it has six squares for its sides.
congruent rectangle bases lying in parallel planes (one directly aligned with the other) with corresponding vertices joined, making four rectangular lateral faces.
Congruent polygon bases lying on parallel planes with corresponding vertice sjoined, makign parallelogram lateral faces. (prisms can be slanted)
polygon base with each vertex joined to a point (called the vertex) on a different plane, making triangular lateral faces. (pyramids can be slanted).
congruent circular bases lying on parallel planes joined by one lateral face (cylinders can be slanted).
a circular base joined to a vertex on a different plane, making one curved lateral face (cones can be slanted).
The set of all points in space which lie a certain distance (called the radius) from a given points. (called the center).
- Volume: Bh where B is area of base
- Surface Area: 2B + Ph, where P is perimeter of base
- Volume: 1/3Bh
- Surface Area: B plus areas of triangle sides
- Volume: pie ( r^2) h
- Surface Area: 2pie (r^2) + 2 pie (r) (h)
- Volume: 1/3 pie (r^2) h
- Surface Area: pie (r^2) + pie r l where l is slant height
- Volume: 4/3 pie (r^3)
- Surface Area: 4 pie (r^2)
What did archimedes found?
he found that the volume of a sphere by showing that the ratio of the volume of a sphere to teh volume of its circumscribing cylinder is 2:3. He was so proud of this result that he reportedly it put on his tombstone.
archimeds noticed that if you take a double cone, a sphere, and a cylinder all of the same radius r and height 2r and if you take the circular slices through all three of them at teh same height, then the area of cylinder slice is the sum of the areas of the double cone slice and the sphere slice.
the volume of the cylinder is the sume of the volume of teh sphere and the voluen fo the cone.
formula looks like: pie (r^2) (2r) = V(sphere) + 2(1/3 pie r^2 r)
How are surface areas found?
Surface of a prism or a pyramid are found by adding up the areas of their polygonal faces. If the pyramid is nto slanted, then it's slant height L (the height of each triangular face) is constant.
Cylinder Surface Areas
For a cylinder, the surface has three faces, two circular bases plus on lateral side. If you think of a soup can label. this lateral side can be unrolled into a rectangle whose dimensions are the circumference of the base (2 pie r) and the height h. So the surface area of a cylinder is 2pie r^2 + 2 pie r h
- For a cone if we use an analogous argument as in the pyramid case, we find that its surface area is
- pie r^2 + 1/2 P L = pie r^2 +pie r l because the perimeter P equals 2 pie r the circumference of the base.
- Calculus can be used to find its surface area somewhat directly although archimedes knew the forumla.
- One can project the sphere onto the lateral side of its circumscribing cylinder (like projecting the surface of earth onto a rectangular map) in such a way that areas are preserved. The lateral side of the cylinder of a radius r and height 2r has area (2pie r ) (2r) = 4 pie r is the surface area of a sphere of radius r.
function that maps points on the plane to other points on the plane (or in space) in such a way that all distances are preserved. so if we call our isometry F, then the distance between points A and B is equal to the distance between F(A) and F(B). A common notation is to use primes for the transformed points. So, under an isometry A'B' = AB. The most common isometries are rotations, reflection in a line ( or in a plane) and translations.
Basic Properties of Isometries
because isometries preserve lengths, they also preserve any geometric property that follows from lengths, such as congruence of triangles, and thus of angles, polygons, and solids, too. areas and volumes are preserved by isometries.
is a function that maps the points of the plane ( or space) to other points of the plane (or space) in such a way that the lengths between any two points are multiplied by a constant value.
Similarity theorem: because corresponding angles of similar triangles are congruent.
is a partial sequence of elements from a set. The order of that sequence is important.
Permutation often use multiplication. To count the number of ways to pick first, second, and third place from 10 students, we start by seeing htat there are 10 ways to choose the first place student. After that, there are 9 contestants from which to choose teh second place student. Then, there are only 8 left from which to choose the third palce student. So there are (10)(9)(8) = 720
is a selection of a certain number of elements from aset. The order in which the selection is not important. The number of ways to chose k objects from a set of n objects denoted n C k (read "n choose k") or (n.. k ) and is equal to n! / k! (n-k)!
if we were to count the number of ways to pick a committee of 3 students from a class of 10, we were not interested in ranking them. We could still count them using the raning above, but then we are overcountibg because sometimes different ranking will be made up of the same three people. In fact there are 3! = 6 ways to rank three people, and so we have overcounted the number of combinations by factor of 6. Therefore, there are only 720/6 = 120 = 10 C 3 possible committee of three students from class of ten.
Basic Rules of Counting
- there are a few axioms:
- "if there are n total outcomes if k of those outcomes meet condition A, then the number of outcomes than do not meet the condition a (n-k)
"if two choices are independent of each other, and there are m ways to make the first choice and n ways to mae the second choice, then the total number of ways to make both choices mn"
the total number of events that meet condition a or condition B is the total number that meet A plus the total number that meet B minus the number that meet both A and bB (due to overcounting.)
finite probability measures the probability of certain outcomes when there are only a finite number of outcomes possible. in finite probability, if all possible outcomes are equally likely, then the probability of an outcome A is the ratio of the number of outcomes in which A occurs to the total number of outcomes.
the probability of event A is P(A) then the probability that A does not happen (A) called the complement of a is P(A) = 1- P(A)
Independent of each other
(A and B) are independent of each other, then the probability of both occurring (A and B) is P (A upside down U B) = P(A) P(B) this is a called the multiplication Rule.
A and B are two events: probability of either one of them happening
(A or B) is P ( A U B) = P(A) + P(B) - P(A upside U B )
where as in the counting case, we subtract the probability of both A and B happening due to having counting case, we subtract the probability of both A and B happening due to having counting it twice.
Probability: (A OR B) is P(A U B) = P(A) + P(B). this is called the addition rule
Probability as a ratio
the first part of the ratio is the number of events in which a desired outcomes occurs, while the second part of the ratio is the total number of events. Probability can also be interpreted as teh likelihood of a given event occurring on a given trial.
it is a very important in statistics as we will see later. for the purposes of probability, teh normal curve is centered at teh mean value of your variable and the standard deviation is the horizontal distance from the center to the steepest point on the curve (inflection point)
when you have several independent trials each with teh same probability of "success" such as flipping a coin several times, or rollign a die several times. here we use the binomial distribution to count B(n,k,p) the probability of having k sucesses in n trials, if each trial's probaility of success is p.
Suppose there is na event that happens randomly but at a constant average rate. Then the time between events is a random variable that is exponentially distributed. For example, if you are studying a radioactive element, and you are measuring the time between two decay events, you would find that the time between events follow the exponential distribution. the time you have to wait until your next phone call might also be exponentially distributed (assuming that phone calls occur at a constant average rate)