Geo U10 Vocab

• arc length - The length of an arc (in inches, centimeters, feet, etc.) is the distance from one of the arc’s endpoints to the arc’s other endpoint, measured around the circle.  Note that arc length is different from arc measure.  Arcs of two different circles may have the same arc measure (like 90∘), but have different arc lengths if one circle has a larger radius than the other one. 

  
• chord - A line segment with its endpoints on a circle.  A chord that passes through the center of a circle is called a diameter. 
• conditional probability - The probability of outcome A occurring, given that outcome B has already happened, is called the conditional probability of A given B.  One way to calculate the conditional probability of A given B is to use the Multiplication Rule, P(A given B) = P(A and B)/P(B).  In many situations it is possible to calculate the conditional probability directly from the data by counting the number of outcomes (or computing the probabilities) of A that are within the outcomes (probabilities) for B.  
• factorial - A shorthand notation for the product of a list of consecutive positive integers from the given number down to 1:  For example, 5!=5·4·3·2·1=120.
• inscribed angle - An angle with its vertex on the circle and sides intersecting the circle at two distinct points.
• inscribed angle theorem - The measure of an inscribed angle is half the measure of its intercepted arc.  Likewise, the measure of an intercepted arc is twice the measure of an inscribed angle whose sides pass through the endpoints of the arc.  In the figure below, if mAC⏜=60°, then m<AMC = 30°.
• major arc - An arc with measure greater than 180°.  Each major arc has a corresponding minor arc that has a measure that is less than 180°.  A major arc is named with three letters on the arc. 
• mutually exclusive - Two events {A} and {B} are mutually exclusive if they have no outcomes in common.  That is, P(A and B) = 0.  This situation is also called disjoint.
• secant - A line that intersects a circle at two distinct points.
• tangent - A line on the same flat surface as a circle that intersects the circle in exactly one point.  A tangent of a circle is perpendicular to a radius of the circle at their point of intersection (also called the point of tangency).
• zero factorial - Zero factorial is 1, 0!=1. 
• circumscribed circle - A circle circumscribes a polygon when it passes through all of the vertices of the polygon.
• Decision chart


  A method for counting the number of outcomes (the size of the sample space) for a sequence of probabilistic situations, often in which the order of the items matters
• A tree diagram could be used to count the number of outcomes, but if the number of outcomes is large a decision chart is more useful
• Example of using a decision chart

  • How many three-letter arrangements could be made by lining up any three blocks, chosen from a set of 26 alphabet blocks, if the first letter must be a vowel and the blocks are not reused?
• Steps in the decision chart

  1. 3 decisions (3 blocks to be chosen)

  2. 5 choices for the first letter (a vowel)

  3. 25 choices for the second letter

  4. 24 choices for the third letter
• The decision chart is a short way to represent a tree with 5 branches for the first alphabet block, followed by 25 branches for each of those branches; each of those 125 branches would then have 24 branches representing the possibilities for the third alphabet block
• fundamental principle of counting - A method for counting the number of outcomes (the size of the sample space) of a probabilistic situation, often in which the order of the items matters.  If event {A} has m outcomes, and event {B} has n outcomes after event {A} has occurred, then event {A} followed by event {B} has m*n  outcomes.  For a sequence of events, a tree diagram could be used to count the number of outcomes, but if the number of outcomes is large a decision chart is more useful.
• permutation - An arrangement of items in which items cannot be selected more than once and the order of selection matters.  For example, if each of five letters, A, B, C, D, E, is printed on a card, the number of 3-letter sequences can you make by selecting three of the five cards is a permutation, 5P3=5!/2!=5⋅4⋅3⋅2/2⋅1=5·4·3=60 .
• radius (of a circle) - Of a circle: A line segment drawn from the center of a circle to a point on the circle.  Note that the length of a circle’s radius is half the length of the circle’s diameter.
• radius (of a polygon) - Of a regular polygon: A line segment that connects the center of a regular polygon with a vertex.  The length of a radius is usually denoted r.
• two-way table - A way to display categorical two-variable data.  The categories of one of the variables is the header of the rows, the other variable is the header of the columns.  The entries in the table can be counts (frequencies) or percents (relative frequencies). 
• association - A relationship between two (or more) variables.  An association between numerical variables can be displayed on a scatterplot, and described by its form, direction, strength, and outliers.  Possible association between two categorical variables can be studied in a relative frequency table.
• central angle - An angle with its vertex at the center of a circle.
• circumcenter - The center of the circle that passes through the vertices of a triangle.  It can be found by locating the point of intersection of the perpendicular bisectors of the sides of the triangle.  The other points of concurrency studied in this course are the centroid and the incenter.
• combination - A combination is an arrangement of items that were selected without repeating items and without regard to order.  The number of combinations that can be made by selecting r items from a set of n items can be found by dividing the total number of permutations by the number of arrangements of a single permutation, nCr=nPr/r!, which can be rewritten as nCr=nPr/r!=n!/r!(n-r). 
• diameter - A line segment drawn through the center of a circle with both endpoints on the circle.  The length of a diameter is usually denoted d.  Note that the length of the diameter of a circle is twice the length of its radius.
• Independent events


  • Rolling a normal six-sided die twice and wanting to know the probability of rolling a 1 twice. The result of the first roll does not affect the probability of rolling a 1 on the second roll.
• P(A given B)


  Probability of A given B
• If two events are independent


  P(A and B) = P(A) · P(B)
• intercepted arc - The arc of a circle bounded by the points where the two sides of an inscribed angle meet the circle.  In the circle below, ∠ABC intercepts AC⏜. 
• sample space - The total number of outcomes that can happen in an event.  The sample space for rolling a standard 6-sided die is the set {1, 2, 3, 4, 5, 6} because those are the only possible outcomes.  
• x^2 + y^2 = r^2
• Addition rule for probability - For any two events {A} and {B},  P(A or B) = P(A) + P(B) - P(A and B) 
• The number of permutations that can be made by selecting r items from a set of n total items can be represented with tree diagrams or decision charts, or calculated using the formula nPr=n!/(n-r)!