Given these four points: $A(3,-10), B(-1,10), C(-8,-6), D(-10,-3)$, find the coordinates of the midpoint of line segments $A B$ and $C D$. Round decimals to the nearest tenth.
midpoint of $A B(x, y)=($
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midpoint of $C D(x, y)=($
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Answer (1)
Calculate the midpoint of line segment A B using the midpoint formula: M A B = ( 2 3 + ( − 1 ) , 2 − 10 + 10 ) = ( 1 , 0 ) .
Calculate the midpoint of line segment C D using the midpoint formula: M C D = ( 2 − 8 + ( − 10 ) , 2 − 6 + ( − 3 ) ) = ( − 9 , − 4.5 ) .
The midpoint of A B is ( 1 , 0 ) .
The midpoint of C D is ( − 9 , − 4.5 ) .
( 1 , 0 ) and ( − 9 , − 4.5 )
Explanation
Problem Analysis and Setup We are given four points: A ( 3 , − 10 ) , B ( − 1 , 10 ) , C ( − 8 , − 6 ) , D ( − 10 , − 3 ) . Our goal is to find the midpoints of line segments A B and C D . We will use the midpoint formula, which states that the midpoint of a line segment with endpoints ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Midpoint of AB First, let's find the midpoint of line segment A B . The coordinates of A are ( 3 , − 10 ) and the coordinates of B are ( − 1 , 10 ) . Applying the midpoint formula, we have: M A B = ( 2 3 + ( − 1 ) , 2 − 10 + 10 ) M A B = ( 2 2 , 2 0 ) M A B = ( 1 , 0 ) So, the midpoint of A B is ( 1 , 0 ) .
Midpoint of CD Next, let's find the midpoint of line segment C D . The coordinates of C are ( − 8 , − 6 ) and the coordinates of D are ( − 10 , − 3 ) . Applying the midpoint formula, we have: M C D = ( 2 − 8 + ( − 10 ) , 2 − 6 + ( − 3 ) ) M C D = ( 2 − 18 , 2 − 9 ) M C D = ( − 9 , − 4.5 ) So, the midpoint of C D is ( − 9 , − 4.5 ) .
Final Answer Therefore, the midpoint of line segment A B is ( 1 , 0 ) and the midpoint of line segment C D is ( − 9 , − 4.5 ) .
Examples
In urban planning, finding the midpoint between two locations can help determine the optimal placement for a new facility, such as a park or community center, to ensure it is equally accessible to residents in both areas. For instance, if we consider two residential areas represented by points A and B, the midpoint of the line segment AB would be the ideal location for the facility. Similarly, in logistics, the midpoint between two warehouses can be the best location for a distribution hub.
