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1/8/13 Unit 1 Basics of Geometry Segments and Rays 1 Segment Definition: Part of a line that consists of two points called the endpoints and all points between them. How to sketch: How to name: A B AB or BA The symbol AB is read as "segment AB". AB (without a symbol) means the length of the segment or the distance between points A and B. 2 Distance on a number line : Find the distance between P and K. G H I J K L M N O -5 Note: P Q R S 5 The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Therefore, the coordinates of points P and K are 3 and -2 respectively. Substituting the coordinates in the formula │a – b │ PK = | 3 - -2 | = 5 Remember : Distance is always positive 4 Between Definition: X is between A and B if AX + XB = AB. X A X AX + XB = AB B A B AX + XB > AB 5 The Segment Addition Postulate Postulate: If C is between A and B, then AC + CB = AB. Example: If AC = x , CB = 2x and AB = 12, then, find x, AC and CB. B 2x A x C Step 1: Draw a figure Step 2: Label fig. with given info. Step 3: Write an equation Step 4: Solve and find all the answers 12 AC + CB = AB x + 2x = 12 3x = 12 x = 4 x = 4 AC = 4 CB = 8 6 Congruent Segments Definition: Segments with equal lengths. (congruent symbol: ) B Congruent segments can be marked with dashes. A If numbers are equal the objects are congruent. C D AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Correct notation: AB = CD AB CD Incorrect notation: AB CD AB = CD 7 Midpoint Definition: A point that divides a segment into two congruent segments If DE EF , then E is the midpoint of DF. Formulas: F E D On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is a b . 2 In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates ( x1 , y1 ) and ( x2 , y2 ) is x1 x2 y1 y2 , 2 2 . 8 Midpoint on Number Line - Example Find the coordinate of the midpoint of the segment PK. G H I J K L M N O P Q -5 R S 5 a b 3 (2) 1 0.5 2 2 2 Now find the midpoint on the number line. 9 Segment Bisector Definition: Any segment, line or plane that divides a segment into two congruent parts is called segment bisector. A F A B E AB bisects DF. B D E AB bisects DF. F A E D D F Plane M bisects DF. B AB bisects DF. 10 Ray Definition: RA : RA and all points Y such that A is between R and Y. A How to sketch: R A Y R How to name: RA ( not AR ) RA or RY ( not RAY ) ( the symbol RA is read as “ray RA” ) 11 Opposite Rays Definition: If A is between X and Y, AX and AY are opposite rays. ( Opposite rays must have the same “endpoint” ) X A Y opposite rays not opposite rays D E DE and ED are not opposite rays. 12 Geometric Concepts • Through any two points there is exactly one line. • A line contains at least two points. • Through any three points, there is exactly one plane. • A plane contains at least three points. 13 • If two planes intersect, then the intersecting is a line. • If two points lie in a plane, then the line containing the two points lie in the same plane. 14