Find the coordinates of M, the midpoint of $\overline{UV}$, for $U (-4,2)$ and $V (6,10)$.
M(-5,-4)
M(-1,8)
M(1,6)
M(2,12)
Answer (2)
The problem asks to find the midpoint M of a line segment U V given the coordinates of the endpoints U and V .
The midpoint formula M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) is used.
Substitute the coordinates of U ( − 4 , 2 ) and V ( 6 , 10 ) into the formula.
Calculate the coordinates of the midpoint: M ( 1 , 6 ) .
Explanation
Problem Analysis and Strategy We are given two points, U ( − 4 , 2 ) and V ( 6 , 10 ) , and we want to find the midpoint M of the line segment U V . The midpoint formula is a straightforward way to calculate the coordinates of the midpoint given the coordinates of the endpoints.
State the Midpoint Formula The midpoint formula is given by: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two endpoints. In our case, U ( − 4 , 2 ) and V ( 6 , 10 ) , so x 1 = − 4 , y 1 = 2 , x 2 = 6 , and y 2 = 10 .
Apply the Formula and Calculate Now, we substitute the coordinates of U and V into the midpoint formula: M = ( 2 − 4 + 6 , 2 2 + 10 ) M = ( 2 2 , 2 12 ) M = ( 1 , 6 ) So, the coordinates of the midpoint M are ( 1 , 6 ) .
Final Answer Therefore, the midpoint M of the line segment U V with endpoints U ( − 4 , 2 ) and V ( 6 , 10 ) is M ( 1 , 6 ) .
Examples
The midpoint formula is useful in various real-world scenarios. For example, if you and a friend want to meet at a location that is exactly halfway between your houses, you can use the midpoint formula to find the coordinates of that meeting point on a map. Similarly, in computer graphics, the midpoint formula is used to find the center of a line segment, which is essential for drawing and manipulating shapes.
The midpoint of the line segment connecting points U(-4, 2) and V(6, 10) is found using the formula, resulting in midpoint coordinates M(1, 6).
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