Midpoint

The point halfway between two points on the coordinate plane is found by calculating the average values of the x and y coordinates.

The mid-point formula is:  \left ( \frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right ) 

Examples:

Use the distance formula and find the distance between two points and use the mid-point formula to find the point halfway between the same two points.

1.  \left ( 1,1 \right )\; and\; \left ( 9.7 \right )

d = \sqrt{\left ( 9-1 \right )^{2}+\left ( 7-1 \right )^{2}}                   mid-pt. = \left ( \frac{1+9}{2} ,\frac{1+7}{2}\right )

   = \sqrt{8^{2}+6^{2}}                                                   = \left ( \frac{10}{2},\frac{8}{2} \right ) 

   =\sqrt{64+36}                                                   = \left ( 5,4 \right ) 

  =\sqrt{100} = 10 

2.  \left ( -4,10 \right )\; and\; \left ( 4,-5 \right ) 

d = \sqrt{\left ( -4-4 \right )^{2}+\left ( 10+5 \right )^{2}}                mid-pt. = \left ( \frac{-4+4}{2},\frac{10-5}{2} \right ) 

   = \sqrt{\left ( -8 \right )^{2}+\left ( 15 \right )^{2}}                                          = \left ( \frac{0}{2} ,\frac{5}{2}\right ) 

   = \sqrt{64+225}                                                  = \left ( 0,\frac{5}{2} \right )\; or\; \left ( 0,2.5 \right ) 

   = \sqrt{289}  = 17  

 

Filed under: Pre-requisites for Pre-Calculus

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