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.  

Find the midpoints of line segments AB and CD. 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 y-coordinates. -30 -20 -10 10 20 30 a b

Midpoint Coordinates = 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 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 1 2 3 4 5 6

Example 1 - Find a Midpoint on a Number Line Q R -6 -5 -4 -3 -2 -1 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 (x1, y1) and Q be (x2, y2)

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 (x2, y2) 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 First, let’s try to visualize the problem. This will give you an idea of where X is located. Q Y

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