What is the midpoint M given $Q (-2,-4)$ and $R (6,10)$?
A. $(-4,-7)$
B. $(1,-2)$
C. $(2,3)$
D. $(-3,8)$
Answer (1)
The midpoint formula is M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Substitute the coordinates of the given points Q ( − 2 , − 4 ) and R ( 6 , 10 ) into the midpoint formula.
Calculate the x -coordinate of the midpoint: 2 − 2 + 6 = 2 4 = 2 .
Calculate the y -coordinate of the midpoint: 2 − 4 + 10 = 2 6 = 3 .
The midpoint is ( 2 , 3 ) .
Explanation
Problem Analysis and Midpoint Formula We are given two points, Q ( − 2 , − 4 ) and R ( 6 , 10 ) , and we want to find the midpoint M of the line segment QR . The midpoint formula is given by:
M = ( 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 two points.
Substitute Coordinates Now, we substitute the coordinates of points Q and R into the midpoint formula:
M = ( 2 − 2 + 6 , 2 − 4 + 10 )
Simplify Next, we simplify the expression:
M = ( 2 4 , 2 6 )
M = ( 2 , 3 )
Final Answer Therefore, the midpoint M of the line segment QR is ( 2 , 3 ) .
Examples
In real life, the midpoint formula is useful in various scenarios, such as finding the center of a room or a field. For example, if you have a rectangular garden and you want to place a sprinkler in the exact center to ensure even watering, you can use the midpoint formula to find the coordinates of the center point. This ensures that the sprinkler is equidistant from all corners of the garden, optimizing its performance. Another example is in sports, where determining the midpoint of a playing field can be crucial for strategic positioning or marking locations.
