The endpoints of $\overline{A B}$ are $A (13.8,7)$ and $B (6.4,10)$. Find the coordinates of the midpoint.
Answer (2)
Identify the coordinates of the endpoints A and B.
Apply the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Calculate the x-coordinate: 2 13.8 + 6.4 = 10.1 .
Calculate the y-coordinate: 2 7 + 10 = 8.5 . The midpoint is ( 10.1 , 8.5 ) .
Explanation
Problem Analysis We are given the coordinates of the endpoints of a line segment A B as A ( 13.8 , 7 ) and B ( 6.4 , 10 ) . Our goal is to find the coordinates of the midpoint of this line segment.
Midpoint Formula The midpoint formula states that the coordinates of the midpoint M of a line segment with endpoints A ( x 1 , y 1 ) and B ( x 2 , y 2 ) are given by:
M = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
In our case, x 1 = 13.8 , y 1 = 7 , x 2 = 6.4 , and y 2 = 10 .
Calculations Now, we will substitute the given values into the midpoint formula to find the coordinates of the midpoint M .
The x-coordinate of the midpoint is:
x = 2 13.8 + 6.4 = 2 20.2 = 10.1
The y-coordinate of the midpoint is:
y = 2 7 + 10 = 2 17 = 8.5
Therefore, the coordinates of the midpoint M are ( 10.1 , 8.5 ) .
Final Answer The coordinates of the midpoint of the line segment A B with endpoints A ( 13.8 , 7 ) and B ( 6.4 , 10 ) are ( 10.1 , 8.5 ) .
Examples
The midpoint formula is useful in various real-world scenarios. For example, if you want to place a new facility (like a store or a warehouse) exactly halfway between two existing locations to minimize travel distance, you would use the midpoint formula to find the optimal location. Similarly, in computer graphics, the midpoint formula is used to draw lines and curves efficiently by calculating intermediate points.
The midpoint of the line segment with endpoints A ( 13.8 , 7 ) and B ( 6.4 , 10 ) is calculated using the midpoint formula. After performing the calculations for the x and y coordinates, we find the midpoint to be ( 10.1 , 8.5 ) . Therefore, the coordinates of the midpoint are ( 10.1 , 8.5 ) .
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