1.2.2 Midpoint and Distance in the Coordinate Plane (continued)

In this example, a segment is given with endpoints C ( − 3, 4) and D (1, − 2). Use the midpoint formula to find the midpoint of this segment. First identify the values of x 1 , x 2 , y 1 , and y 2 . It doesn’t matter which point, C or D , is picked to be ( x 1 , y 1 ). Here, D (1, − 2) is ( x 1 , y 1 ) and C ( − 3, 4) is ( x 2 , y 2 ). So x 1 = 1, x 2 = − 3, y 1 = − 2, and y 2 = 4. Substitute these values into the midpoint formula and simplify to find the x - and y -coordinates of the midpoint.

Example 2 Finding the Coordinates of an Endpoint

The midpoint formula can be used to find one of the endpoints of a segment when the other endpoint and the midpoint of the segment are known. In this example, the endpoints are A and B , and the midpoint is M . The coordinates of A and M are known, and the coordinates of B are unknown. So, let the coordinates of B be ( x , y ). Now, substitute the coordinates into the midpoint formula where ( x 1 , y 1 ) = (3, 5) and ( x 2 , y 2 ) = ( x , y ). The first coordinate from the formula is equal to the x -coordinate from the midpoint. So, set the first coordinate from the formula equal to 4. Then solve the equation for x , which is the x -coordinate of B . Use a similar process to find the y -coordinate of B .

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