Distance and Midpoints Objective 1To find the distance

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. 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 The coordinate of the midpoint is 2. E is the midpoint.

More Midpoint • For the midpoint on a coordinate plane, the formula is: This is the midpoint. B(-1, 7) A(-8, 1)

Finding the endpoint of a segment • We’re still going to use the Midpoint Formula: Find the coordinates for X if M(5, -1) is the midpoint and the other endpoint has coordinates Y(8, -3) • But there will be a few unknowns: • helps us find the x-coordinate of the endpoint.

Finding the endpoint of a segment Multiply both sides by 2 to eliminate the denominator -8 x 2 = 2 This helps us find the y-coordinate of the midpoint -8 Subtract 8 from both sides This is the x-coordinate of the other endpoint

Finding the endpoint of a segment +3 y 2 = 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 3 x – 5 = x + 9 -x -x 2 x – 5 = 9 +5 +5 2 x = 14 A 3 x - 5 M x+9 2 x = 14 2 x=7 2 B

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)