Point $M$ is the midpoint of segment KL. Point $N$ is the midpoint of segment ML.
Point $K$ is located at $(-7,-6)$, and point $L$ is located at $(1,10)$. What are the coordinates of point $N$?
A. $(-1,6)$
B. $(-2,6)$
Answer (2)
Find the midpoint M of segment KL using the midpoint formula: M = ( 2 K x + L x , 2 K y + L y ) .
Calculate the coordinates of M: M = ( 2 − 7 + 1 , 2 − 6 + 10 ) = ( − 3 , 2 ) .
Find the midpoint N of segment ML using the midpoint formula: N = ( 2 M x + L x , 2 M y + L y ) .
Calculate the coordinates of N: N = ( 2 − 3 + 1 , 2 2 + 10 ) = ( − 1 , 6 ) . The coordinates of point N are ( − 1 , 6 ) .
Explanation
Find the coordinates of point M First, we need to find the coordinates of point M, which is the midpoint of segment KL. The midpoint formula is given by:
M = ( 2 K x + L x , 2 K y + L y )
where K x and K y are the x and y coordinates of point K, and L x and L y are the x and y coordinates of point L.
Given K = ( − 7 , − 6 ) and L = ( 1 , 10 ) , we can substitute these values into the midpoint formula to find the coordinates of point M:
M = ( 2 − 7 + 1 , 2 − 6 + 10 ) M = ( 2 − 6 , 2 4 ) M = ( − 3 , 2 )
Find the coordinates of point N Next, we need to find the coordinates of point N, which is the midpoint of segment ML. Using the midpoint formula again:
N = ( 2 M x + L x , 2 M y + L y )
where M x and M y are the x and y coordinates of point M, and L x and L y are the x and y coordinates of point L.
Given M = ( − 3 , 2 ) and L = ( 1 , 10 ) , we can substitute these values into the midpoint formula to find the coordinates of point N:
N = ( 2 − 3 + 1 , 2 2 + 10 ) N = ( 2 − 2 , 2 12 ) N = ( − 1 , 6 )
Final Answer Therefore, the coordinates of point N are ( − 1 , 6 ) .
Examples
In computer graphics, finding midpoints is crucial for various tasks such as drawing lines, curves, and shapes. For instance, if you want to draw a line between two points and then find the middle point of that line to place an object, you would use the midpoint formula. This concept is also used in pathfinding algorithms, where finding the midpoint between two nodes can help determine the shortest path.
The coordinates of point N are ( − 1 , 6 ) . This is found by calculating the midpoints of segments KL and ML step-by-step using the midpoint formula. Therefore, option A is the correct choice.
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