Explore: The Midpoint Formula Solutions

The Midpoint Formula Solutions

The midpoint formula is used to find the point equidistant from two endpoints on the coordinate plane. 
The midpoint formula is: $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$  
The midpoint formula calculates the average of the x-values and the average of the y-values written as an ordered pair. 
If you have the midpoint and an endpoint, you can determine an unknown endpoint by working backward.  
Substitute all known values into the formula and solve for the unknown values.

Example 3

Calculate the midpoint between points CD. Mark the midpoint, M, on the graph. 
Find the endpoint, E, with endpoint M and point D as the midpoint.

$C (4, 6), D (- 3, - 2) (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) (\frac{4 + - 3}{2}, \frac{6 + - 2}{2})$

$(\frac{1}{2}, \frac{4}{2}) M (0.5, 2)$

$M (0.5, 2), D (- 3, - 2) (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) (\frac{x + 0.5}{2}, \frac{y + 2}{2}) = (- 3, - 2)$

$\begin{matrix} \frac{x + 0.5}{2} = - 3 & \frac{y + 2}{2} = - 2 \\ x + 0.5 = - 6 & y + 2 = - 4 \\ x = - 6.5 & y = - 6 \end{matrix}$

Example 4

Mini’s Mini Golf Course has a putting green for golfers to practice on before their games. Currently, there are three tees and two holes on the green. The plan is to add another hole between the existing holes.

Find the midpoint between the existing holes.

$(- 3, 8), (3, 0) (\frac{- 3 + 3}{2}, \frac{8 + 0}{2}) (0, 4)$

Determine the midpoint, horizontal and vertical, for the entire course from the middle tee (#2). Explain.

Horizontal:

The y-axis is halfway between the left and right side walls of the green. Because that is the dividing line, no calculation is needed.

Vertical:

$(0, - 6), (0, 10) (\frac{0 + 0}{2}, \frac{- 6 + 10}{2}) (0, 2)$

Determine the distance in feet from the middle of the putting green to the new hole from part A. Explain.

The distance is 2 feet. The points $(0, 4)$ to $(0, 2)$ align vertically, so you can count rather than use the distance formula.