The point halfway between the endpoints of a

The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal parts.

In Coordinate Geometry, there is more than one way to determine the midpoint of a line segment.

Method 1 You may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints. Find the midpoints of line segments AB and CD. The length of line segment AB is 8 (by counting). The midpoint is 4 units from either endpoint. On the graph, this point is (1, 4). The length of line segment CD is 3 (by counting). The midpoint is 1. 5 units from either endpoint. On the graph, this point is (2, 1. 5)

Method 2 If the line segments are diagonal, more thought must be paid to the solution. When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the ycoordinates. -30 -20 -10 a 0 10 20 30 b

Midpoint Formula Midpoint Coordinates = The Midpoint Formula works for all line segments: vertical, horizontal or diagonal.

Example 1 - Find a Midpoint on a Number Line Q R -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Method 1 The distance from -3 to 4 is 7. Half of 7 is 3. 5, which you add to -3. The midpoint is 0. 5. +7 +3. 5 Q R -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Example 1 - Find a Midpoint on a Number Line Q R -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Method 2 Use the midpoint formula

Example 2 – Find the Coordinates of a Midpoint TEMPERATURE Find the coordinate of the midpoint PQ The coordinates of P and Q are -20 and 40. Let M be the midpoint of PQ Q = 40 P = -20 Find the coordinates of M, the midpoint of PQ, for P( -1, 2) and Q(6, 1) Let P be (x 1, y 1) and Q be (x 2, y 2)

Example 3 - A Different Problem Find the Coordinates of an Endpoint Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8) Method 2 Let Z be (x 2, y 2) in the Midpoint Formula Write two equations to find the coordinates of x

A Different Approach to a Different Problem Find the Coordinates of an Endpoint Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8) Method 1 Q First, let’s try to visualize the problem. This will give you an idea of where X is located. Y

Example 3 – Use Algebra to Find Measures A is the midpoint of segment BC. Find x and the measures of AB and AC. 2 x + 14 C 3 x + 6 A B

Exercises Given the endpoints of a line segment, find the midpoint 1) (7 , 4), (9, − 1) 2) (8, − 9), (0, 5) 3) (1, − 7), (1, − 12) 4) (0, 4), (− 4, -12) 5) (− 4, 2), (2, − 3) 6) (5, 9), (− 1, 9) 7) (− 7, 8), (− 2, − 9) 8) (2, − 11), (− 9, 0) 9) (4, − 1), (2, − 7) 10) (− 4, − 6), (3, − 6) 11) (14, 0), (− 7, 5) 12) (14, − 8), (12, − 1) 13) (− 4, 12), (− 7, − 2)

Given the midpoint and one endpoint of a line segment, find the other endpoint. 1) Endpoint: (− 9, − 1), midpoint: (8, 14) 2) Endpoint: (10, 12), midpoint: (6, 9) 3) Endpoint: (− 8, − 10), midpoint: (10, − 7) 4) Endpoint: (− 11, 9), midpoint: (3, − 11) 5) Endpoint: (− 2, 7), midpoint: (12, − 10) 6) Endpoint: (11, 14), midpoint: (10, 14) 7) Endpoint: (14, − 8), midpoint: (5, 8) 8) Endpoint: (− 9, 0), midpoint: (10, − 7)