The Midpoint Formula

The Midpoint Formula exercise

• Determine the midpoint.

  Hints

  Just look at the example beside if both points have the same $x$-coordinate.

  The $y$-coordinate is the arithmetic middle of $-2$ and $4$, namely $1$.

  For the given points on the left the midpoint is $(-3,14.5)$.

  You get the $y$-coordinate as follows:

  $14.5=\frac{12+17}{2}=\frac{29}2$.

  In general the midpoint formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$

  Solution

  The first look out point is $(11,8)$ and the second one $(-3,12)$.

  Use the midpoint formula with the coordinates of the given points:

  • We sum the $x$-coordinates and divide the sum by $2$. So we get $\frac{11-3}2=\frac82=4$.
  • We proceed in the same fashion with the $y$-coordinates to get $\frac{8+12}2=\frac{20}2=10$.

  Thus we get the midpoint $(4,10)$.

• Calculate the three points.

  Hints

  The midpoint formula for two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ is given by

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$.

  You have to use this midpoint formula three times.

  • Once for the midpoint $M$ between $P_1$ and $P_2$.
  • Next for the midpoint between $P_1$ and $M$.
  • Lastly for the midpoint between $M$ and $P_2$.

  The coordinates of $M$ are whole numbers.

  Solution

  Imke wants to divide the path starting with $P_1=(1,2)$ and ending with $P_2=(11,8)$ into three equidistant points. So first we have to find the midpoint $M$ of those two points and after two more midpoints: one between the starting point $P_1$ and the midpoint $M$ and the other between the midpoint $M$ and the ending point $P_2$.

  Let's start with the midpoint between $P_1$ and $P_2$.

  We use the midpoint formula,

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$,

  and put the coordinates in

  $M=\left(\frac{1+11}2,\frac{2+8}2\right)=\left(\frac{12}2,\frac{10}2\right)=\left(6,5\right)$.

  Next we use the midpoint formula again for the starting point and $M=(6,5)$:

  $\left(\frac{1+6}2,\frac{2+5}2\right)=\left(\frac{7}2,\frac{7}2\right)=\left(3.5,3.5\right)$.

  And again for $M=(6,5)$ and the ending point:

  $\left(\frac{6+11}2,\frac{5+8}2\right)=\left(\frac{17}2,\frac{13}2\right)=\left(8.5,6.5\right)$.

• Find the midpoints of the given points.

  Hints

  In general the midpoint of two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ is given by the midpoint formula:

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$.

  Pay attention to the sign.

  If two points have a coordinate in common, then that coordinate stays the same for the midpoint.

  The order of adding the coordinates doesn't matter at all.

  Solution

  The midpoint formula for two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ is given by

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$.

  Let's use this formula for a few examples:

  For $P_1=(-3,2)$ and $P_2=(2,-3)$, we get the midpoint

  $M=\left(\frac{-3+2}2,\frac{2+(-3)}2\right)=\left(\frac{-1}2,\frac{-1}2\right)=(-0.5,-0.5)$.

  The points $P_1=(2,2)$ and $P_2=(2,4)$, we get the midpoint

  $M=\left(\frac{2+2}2,\frac{2+4}2\right)=\left(\frac{4}2,\frac{6}2\right)=(2,3)$.

  The midpoint of $P_1=(2,-4)$ and $P_2=(6,8)$ is given by

  $M=\left(\frac{2+6}2,\frac{-4+8}2\right)=\left(\frac{8}2,\frac{4}2\right)=(4,2)$.

  The midpoint of the last two points $P_1=(3,3)$ and $P_2=(-7,9)$ is given by

  $M=\left(\frac{3+(-7)}2,\frac{3+9}2\right)=\left(\frac{-4}2,\frac{12}2\right)=(-2,6)$.

• Calculate the meeting point of Imke and her friend.

  Hints

  You have to add the coordinates and divide by $2$.

  The $x$-coordinate is a rational number while the $y$-coordinate is a natural number.

  Here you see the calculation of the midpoint of $(3,7)$ and $(7,8)$.

  Solution

  Here we have to determine the midpoint of $(2,7)$ and $(9,15)$. To do this, we use the midpoint formula. This means to get a coordinate of the midpoint we add the corresponding coordinates of those points and divide the resulting sum by $2$:

  $M=\left(\frac{2+9}2,\frac{7+15}2\right)=\left(\frac{11}2,\frac{22}2\right)=(5.5,11)$.

• Identify the midpoint formula.

  Hints

  If both points have the same $x$- or $y$-coordinate, you get the midpoint as follows:

  • The common coordinate stays the same for the midpoint.
  • The coordinate not in common is the arithmetic middle of the given (two) coordinates.

  For example, the midpoint of $(4,4)$ and $(6,10)$ is $(5,7)$.

  Solution

  The midpoint of two given points, $(x_1,y_1)$ and $(x_2,y_2)$, is given by the midpoint formula:

  $\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)$.

  To get the $x$-coordinate of the midpoint, add the $x$-coordinates of the two points and divide the sum by $2$. To get the $y$-coordinate, proceed in an similar fashion with the $y$-coordinates of the two points.

• Find the midpoints of the given points.

  Hints

  You can calculate the midpoint as follows:

  • For the $x$-coordinate, add the $x$-coordinates of the points and divide the sum by $2$.
  • Proceed in a similar fashion for the $y$-coordinate.

  Keep in mind that only three points are midpoints.

  The midpoint of $(-2,3)$ and $(4,3)$ is $M=(1,3)$.

  Solution

  We use the midpoint formula for each pair of points: we add the coordinates and divide the sum by $2$:

  Let's start with $A$ and $B$:

  $M=\left(\frac{3+7}2,\frac{1+3}2\right)=\left(\frac{10}2,\frac{4}2\right)=(5,2)$.

  For $A$ and $C$ we get the midpoint

  $M=\left(\frac{3+1}2,\frac{1+5}2\right)=\left(\frac{4}2,\frac{6}2\right)=(2,3)$.

  Lastly we determine the midpoint of $B$ and $C$:

  $M=\left(\frac{1+7}2,\frac{5+3}2\right)=\left(\frac{8}2,\frac{8}2\right)=(4,4)$.