The Midpoint Formula

Want practice with preliminary concepts first?

• Locating Points in Quadrants and on Axes
• Practice with Points

Here, $\,x_1\,$ (read as ‘$\,x\,$ sub $\,1\,$’) denotes the $\,x$-value of the first point, and $\,y_1\,$ (read as ‘$\,y\,$ sub $\,1\,$’) denotes the $\,y$-value of the first point. Similarly, $\,x_2\,$ and $\,y_2\,$ denote the $\,x$-value and $\,y$-value of the second point.

Thus, to find the location that is exactly halfway between two points, you average the $x$-values, and average the $y$-values.

The Midpoint Formula follows easily from the following observations:

• The average of two numbers always lies exactly halfway between the two numbers .
• Referring to the sketch below, $\Delta ABD\,$ is similar to $\,\Delta AMC\,.$ That is, these two triangles have the same angles. Why? They both share angle $\,A\,,$ and they both have a right angle. Since all the angles in a triangle sum to $\,180^\circ\,,$ the third angles must also be the same.

• Similarity gives us what we need! It tells us that $\Delta ABD\,$ and $\Delta AMC\,$ have exactly the same shapes—they're just different sizes. Since $\,\overline{AM}\,$ is exactly half of $\,\overline{AB}\,,$ $\,\overline{AC}\,$ must be exactly half of $\,\overline{AD}\,.$ Thus, $\,C\,$ is the midpoint between $\,A\,$ and $\,D\,$ (which can be found by averaging $\,x_1\,$ and $\,x_2\,$).
• Use a similar argument to show that $\,\overline{DE}\,$ (which has the same length as $\,\overline{CM}\,$) is exactly half of $\,\overline{DB}\,.$

Examples