The park owners want to put an information booth halfway between the roller coaster and the water slide. If the roller coaster is located on the map at $(7,6)$ and the water slide is located at $(-3,-4)$, at which coordinates should the information booth be placed?
A. $(1,1)$
B. $(2,1)$
C. $(2,-1)$
D. $(4,2)
Answer (2)
Identify the coordinates of the roller coaster and the water slide.
Apply the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Substitute the given coordinates into the formula and simplify.
Determine the coordinates of the information booth: ( 2 , 1 ) .
Explanation
Problem Analysis We are given the coordinates of the roller coaster ( 7 , 6 ) and the water slide ( − 3 , − 4 ) . We want to find the coordinates of the information booth, which is located halfway between the roller coaster and the water slide. This means we need to find the midpoint of the line segment connecting these two points.
Midpoint Formula 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.
Applying the Formula Let ( x 1 , y 1 ) = ( 7 , 6 ) and ( x 2 , y 2 ) = ( − 3 , − 4 ) . Substituting these values into the midpoint formula, we get: M = ( 2 7 + ( − 3 ) , 2 6 + ( − 4 ) ) M = ( 2 7 − 3 , 2 6 − 4 ) M = ( 2 4 , 2 2 ) M = ( 2 , 1 ) So, the coordinates of the information booth are ( 2 , 1 ) .
Final Answer Therefore, the information booth should be placed at the coordinates ( 2 , 1 ) .
Examples
In city planning, determining the midpoint between two key locations (like a residential area and a business district) is crucial for placing community centers or public transportation hubs. This ensures equal accessibility for residents from both areas, promoting balanced urban development and efficient resource allocation.
The information booth should be placed at the coordinates ( 2 , 1 ) , which is the midpoint between the roller coaster at ( 7 , 6 ) and the water slide at ( − 3 , − 4 ) . This is calculated using the midpoint formula. Therefore, the correct choice is B. ( 2 , 1 ) .
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