The point $A(-5,3)$ and the point $B$ have midpoint $(4,-2)$. Find the coordinates of $B$.
Answer (2)
Use the midpoint formula to set up equations for the x and y coordinates of point B.
Solve the equation 2 − 5 + x = 4 for x , which gives x = 13 .
Solve the equation 2 3 + y = − 2 for y , which gives y = − 7 .
State the coordinates of point B as ( 13 , − 7 ) .
Explanation
Problem Analysis and Setup We are given point A ( − 5 , 3 ) and the midpoint M ( 4 , − 2 ) of the line segment A B . We need to find the coordinates of point B ( x , y ) . 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, A ( − 5 , 3 ) , B ( x , y ) , and M ( 4 , − 2 ) .
Applying the Midpoint Formula Using the midpoint formula, we have: ( 4 , − 2 ) = ( 2 − 5 + x , 2 3 + y ) This gives us two equations: 2 − 5 + x = 4 2 3 + y = − 2
Solving for x and y Solving for x :
− 5 + x = 2 × 4 − 5 + x = 8 x = 8 + 5 x = 13 Solving for y :
3 + y = 2 × ( − 2 ) 3 + y = − 4 y = − 4 − 3 y = − 7
Final Answer Therefore, the coordinates of point B are ( 13 , − 7 ) .
Examples
The midpoint formula is useful in various real-world scenarios, such as finding the center of a line segment on a map or determining the coordinates of a missing endpoint when you know one endpoint and the midpoint. For instance, if you're designing a bridge and need to ensure it's equidistant from two anchor points, you'd use the midpoint formula to find the exact center point for the bridge's support structure. Another example is in computer graphics, where the midpoint formula is used to calculate the center of a line or shape for transformations and rendering.
The coordinates of point B are (13, -7). This was found using the midpoint formula to solve for both the x and y values of point B given point A and the midpoint. By setting up equations for both coordinates and solving, we determined the position of point B accurately.
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