### Midpoint Formula Worksheets

We all know that to find the midpoint or also known as the average of two real numbers we just simply add the two and then, divide the answer by 2. Let’s take a look in this example; the midpoint of numbers 6 and 10 is 8. By applying the above mentioned rule we can arrive at the solution, $$\left(6+10 \right)\div 2=16\div 2=8$$. In this connection, to get the coordinates of the midpoint of the two given points which can be connected by a line segment (line), we just simply get the average of the x-coordinates of the points as well as the average of the y-coordinates of the points. The following is the general formula in mathematical expression:

$$x=\displaystyle \frac{x_1+x_2}{2} \qquad \qquad \qquad y=\displaystyle \frac{y_1+y_2}{2}$$
In the formula, $$x_1$$ represents the x-coordinate of the first point and $$x_2$$ is the x-coordinate of the second point. It applies the same in y-coordinates.

Examples:

Find the midpoint of each segment with the given endpoints.

1. $$\left(0,3 \right)$$ and $$\left(4,7 \right)$$

Solution:

Substitute the given coordinates correctly and simplify.
\small \begin{align*} x&=\displaystyle \frac{x_1+x_2}{2}\qquad \qquad \qquad &y&=\displaystyle \frac{y_1+y_2}{2} \\ &=\displaystyle \frac{0+4}{2} &&=\displaystyle \frac{3+7}{2} \\&=\displaystyle \frac{4}{2}&&=\displaystyle \frac{10}{2} \\&=2&&=5 \end{align*}
Thus, the coordinates of the midpoint between $$\left(0,3 \right)$$ and $$\left(4,7 \right)$$ is $$\left(2,5 \right)$$.

2. $$\left(-10,4 \right)$$ and $$\left(7,1 \right)$$

Solution:
\small \begin{align*} x&=\displaystyle \frac{x_1+x_2}{2}\qquad \qquad \qquad &y&=\displaystyle \frac{y_1+y_2}{2} \\ &=\displaystyle \frac{-10+7}{2} &&=\displaystyle \frac{4+1}{2} \\&=\displaystyle \frac{-3}{2}&&=\displaystyle \frac{5}{2} \\&=1.5&&=2.5 \end{align*}

The midpoint is $$\left(1.5,2.5 \right)$$ in decimal form or $$\left(\frac{-3}{2},\frac{5}{2} \right)$$ in fractional form.

3. $$\left(6.2,5.8 \right)$$ and $$\left(1.4,6 \right)$$

Solution:
\small \begin{align*} x&=\displaystyle \frac{x_1+x_2}{2}\qquad \qquad \qquad &y&=\displaystyle \frac{y_1+y_2}{2} \\ &=\displaystyle \frac{6.2+1.4}{2} &&=\displaystyle \frac{5.8+6}{2} \\&=\displaystyle \frac{7.6}{2}&&=\displaystyle \frac{11.8}{2} \\&=3.8&&=5.9 \end{align*}

The midpoint is $$\left(3.8,5.9 \right)$$.

Practice Exercises:

Find the midpoint of each of the segment with the given endpoints:
1. $$\left(-3,6 \right)$$ and $$\left(-2,-6 \right)$$
2. $$\left(-9,-3 \right)$$ and $$\left(-1,-5 \right)$$
3. $$\left(11.5,6.9 \right)$$ and $$\left(2.5,8 \right)$$
4. $$\left(-8,-10 \right)$$ and $$\left(15,11 \right)$$