Select the correct answer.
The coordinates of $\overline{ CD }$ are $C (-7,-2)$ and $D (-3,-5)$. Point E is the midpoint of $\overline{ CD }$. What are the coordinates of the midpoint E ?
A. $(-5,-3.5)$
B. $(5,1.5)$
C. (1, 0.5)
D. $(-5,-1.5)$
Answer (1)
The problem asks to find the midpoint E of a line segment CD given the coordinates of the endpoints C and D.
The midpoint formula E = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) is used.
Substitute the coordinates of C(-7, -2) and D(-3, -5) into the formula: E = ( 2 − 7 + ( − 3 ) , 2 − 2 + ( − 5 ) ) .
Simplify to find the coordinates of the midpoint: ( − 5 , − 3.5 ) .
Explanation
Problem Analysis We are given the coordinates of two points, C ( − 7 , − 2 ) and D ( − 3 , − 5 ) , and we want to find the midpoint E of the line segment C D . The midpoint formula is a straightforward way to calculate the coordinates of the midpoint given the coordinates of the endpoints.
Stating the Midpoint Formula The midpoint formula is given by: E = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the endpoints. In our case, C ( − 7 , − 2 ) and D ( − 3 , − 5 ) , so x 1 = − 7 , y 1 = − 2 , x 2 = − 3 , and y 2 = − 5 .
Applying the Formula Now, we substitute the coordinates of points C and D into the midpoint formula: E = ( 2 − 7 + ( − 3 ) , 2 − 2 + ( − 5 ) ) E = ( 2 − 10 , 2 − 7 ) E = ( − 5 , − 3.5 ) So, the coordinates of the midpoint E are ( − 5 , − 3.5 ) .
Final Answer Therefore, the coordinates of the midpoint E of C D are ( − 5 , − 3.5 ) .
Examples
The midpoint formula is useful in various real-world scenarios. For example, if you want to place a new facility (like a cell tower or a warehouse) exactly halfway between two cities to minimize travel distance, you can use the midpoint formula to find the optimal location based on the coordinates of the cities. Similarly, in computer graphics, the midpoint formula helps in drawing lines and curves efficiently by calculating intermediate points.
