The midpoint of $\overline{K L}$ is $M(2,7)$. One endpoint is $K(-5,11)$. Which equations can be used to find the coordinates of endpoint $L \left(x_2, y_2\right)$ ?
$\frac{-5+x_2}{2}=2, \frac{11+y_2}{2}=7$
$\frac{-5+x_2}{2}=7, \frac{11+y_2}{2}=2$
$\frac{2+x_2}{2}=-5, \frac{7+y_2}{2}=11$
$\frac{2+x_2}{2}=11, \frac{7+y_2}{2}=-5
Answer (2)
To find the equations for the coordinates of endpoint L ( x 2 , y 2 ) , we apply the midpoint formula. Given the midpoint M ( 2 , 7 ) and endpoint K ( − 5 , 11 ) , the equations are derived as follows:
Apply the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Substitute the given values: 2 = 2 − 5 + x 2 and 7 = 2 11 + y 2 .
The equations to find the coordinates of endpoint L are: 2 − 5 + x 2 = 2 , 2 11 + y 2 = 7 .
Explanation
Apply the midpoint formula. We are given the midpoint M ( 2 , 7 ) of the line segment K L and one endpoint K ( − 5 , 11 ) . We need to find the equations that can be used to determine the coordinates of the other endpoint L ( x 2 , y 2 ) . The midpoint formula states that the coordinates of the midpoint M of a line segment with endpoints K ( x 1 , y 1 ) and L ( x 2 , y 2 ) are given by
M = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
In our case, K ( x 1 , y 1 ) = K ( − 5 , 11 ) and M ( 2 , 7 ) . Therefore, we have
2 = 2 − 5 + x 2 7 = 2 11 + y 2
These are the equations we need to solve for x 2 and y 2 .
Identify the correct equations. The equations that can be used to find the coordinates of endpoint L ( x 2 , y 2 ) are:
2 − 5 + x 2 = 2 2 11 + y 2 = 7
Comparing these equations with the given options, we find that the correct option is:
2 − 5 + x 2 = 2 , 2 11 + y 2 = 7
Examples
The midpoint formula is useful in various real-world scenarios, such as finding the center of a sports field or determining the location of a meeting point that is equidistant from two individuals' homes. For example, if two friends live at coordinates A ( x 1 , y 1 ) and B ( x 2 , y 2 ) and want to meet at a place exactly in the middle, they can use the midpoint formula to find the coordinates of the meeting point M ( 2 x 1 + x 2 , 2 y 1 + y 2 ) . This ensures a fair and convenient meeting location for both friends.
To find the coordinates of endpoint L ( x 2 , y 2 ) using the midpoint formula, we set up the equations based on the given midpoint M ( 2 , 7 ) and endpoint K ( − 5 , 11 ) . The correct equations are: 2 − 5 + x 2 = 2 and 2 11 + y 2 = 7 . Therefore, the correct choice is: 2 − 5 + x 2 = 2 , 2 11 + y 2 = 7 .
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