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Distance and Midpoints Objective: (1)To find the distance between two points (2) To find the midpoint of a segment Definitions • Midpoint: The points halfway between the endpoints of a segment. • Distance Formula: A formula used to find the distance between two points on a coordinate plane. • Segment Bisector: A segment, line, or plane that intersects a segment at its midpoint. Midpoint • To find the midpoint along the number line, add both numbers and divide by 2. ab 2 A B C D E F G H I J -6 -4 -2 0 2 4 6 8 10 12 Find the midpoint of BH 48 4 2 2 2 The coordinate of the midpoint is 2. E is the midpoint. More Midpoint • For the midpoint on a coordinate plane, the formula is: x1 x2 y1 y2 M , 2 2 B(-1,7) A(-8,1) 8 (1) 1 7 , 2 2 9 8 , 2 2 This is the midpoint. 1 4 ,4 2 Finding the endpoint of a segment • We’re still going to use the Midpoint Formula: x1 x2 y1 y2 M , 2 2 • But the there will be a few unknowns: Find the coordinates for X if M(5,-1) is the midpoint and the other endpoint has coordinates Y(8,-3) x1 x2 • helps us find 2 the x-coordinate of the endpoint. Finding the endpoint of a segment 8 x2 5 2 Multiply both sides by 2 to eliminate the denominator 8 x2 2 25 2 y1 y 2 2 3 y2 1 2 8 x2 10 -8 x2 = 2 -8 Subtract 8 from both sides This is the x-coordinate of the other endpoint This helps us find the y-coordinate of the midpoint 3 y2 2 2 2 1 Finding the endpoint of a segment 3 y2 2 +3 y2 = 1 +3 This is the y-coordinate of the endpoint The coordinate of the other endpoint is X(2,1). Finding the value of a variable M is the midpoint of AB. Find the value of x: Since M is a midpoint, that means that AM=MB which means 3x – 5 = x + 9 -x -x 2x – 5 = 9 +5 +5 2x = 14 A 3x - 5 M x+9 B 2x = 14 2 x=7 2 Distance • Remember: AB means the length of AB To find the distance on the number line, take the absolute value of the difference of the coordinates. a – b A B C D E F G H I J -6 -4 -2 0 2 4 6 8 10 12 Find CJ -2 -12=-14= 14 CJ = 14 Find EA 2 – (-6) =2+6 =8 =8 EA = 8 More Distance The distance between two points in the coordinate plane is found by using the following formula: A(-3,1) B(4,-2) d [4 (3)]2 (2 1) 2 d ( x2 x1 ) 2 ( y2 y1 ) 2 d (7) 2 (3) 2 d 49 9 d 58 d 7.6