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GEOMETRY: CHAPTER 1 CHAPTER 1.3: Segments and Their Measures A postulate or axiom is a rule that is accepted without proof. A theorem is a rule that can be proved. Postulate 1: Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between two points A and B , written as AB , is the absolute value of the difference of the coordinates of A and B. Postulate 2: Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. Congruent Segments are segments that have the same length. In the diagram below, you can say “the length of segment AB is equal to the length of segment CD. You can also say “segment AB is congruent to segment CD.” The symbol means “is congruent to.” Ex. 2: Plot E (1 , 2) , F (5 , 2), G (3 ,4), and H(3 , -1). Then determine whether segment EF is congruent to segment GH. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. Use Ruler Postulate EF 5 1 4 To find the length of a vertical segment, find the absolute value of the difference of the ycoordinates of the endpoints. Use Ruler Postulate GH 4 (1) 5 Segment EF is not congruent with segment GH. The Distance Formula (p.19) Recall from the Pythagorean Theorem that, for a right triangle with hypotenuse of length c and sides of length a and b, you have a2 + b2 = c2 Pythagorean Theorem The Distance Formula (cont.) Suppose you want to determine the distance d between two points (x1,y1) and (x2,y2) in the plane. With these two points, a right triangle can be formed. The distance d between the points (x1,y1) and (x2,y2) in the plane is d = ( x2 x1 )2 ( y2 y1 )2 Ex. 3. Find the distance between the points (-1,3) and (5, -2) ( x2 x1 ) 2 ( y2 y1 ) 2 (5 (1)) (2 3) Step 1—Use the Distance Formula 2 (6) 2 ( 5) 2 2 Step 2—Simplify. 36 25 61 7.81 Step 3—Use a calculator to approximate the value of the square root