Halfway between two points there is one special point: the midpoint. This lesson is about finding it — on a number line, where the midpoint is simply the average of two coordinates, and on the coordinate plane, where the x-coordinates and the y-coordinates are averaged separately. Averaging also runs backward, so one endpoint and the midpoint together reveal the other endpoint.
The midpoint of a segment is the point that splits it into two congruent segments (equal in measure), so it sits exactly halfway between the endpoints. On a number line the midpoint is the average of the two coordinates; on the coordinate plane, average the x-coordinates and the y-coordinates separately. Averaging also runs backward: given one endpoint and the midpoint, the other endpoint is the same distance past the midpoint.
What you are looking at: A segment on a number line. In Find the Midpoint mode the two endpoints are A and B, and M is their midpoint. The single tick mark on each half of the segment shows that the two halves are congruent (equal in measure).
What to do: Drag the orange handle and the purple handle along the number line. The mode buttons under the diagram switch between finding a midpoint and working backward, and the line directly above them explains what each mode lets you drag.
What to notice: The midpoint is the average of the two endpoint coordinates. In Work Backward mode the second endpoint lands the same distance past the midpoint as the first endpoint is on the other side, so the midpoint stays equidistant from both endpoints.
Mode buttons — Find the Midpoint lets you drag endpoint A (orange) and endpoint B (purple), and M is computed as their average. Work Backward lets you drag endpoint A (orange) and the midpoint M (purple), and the other endpoint B is computed so that M stays the midpoint; a drag that would push B off the number line is refused.
What you are looking at: Segment AB on a coordinate grid, with its midpoint M. The single tick mark on each half of the segment shows that the two halves are congruent (equal in measure), and the formula under the grid shows the Midpoint Formula with the current coordinates substituted in, written with stacked fractions.
What to do: Drag the orange handle on A and the purple handle on B to move the endpoints to new grid points. The midpoint, the substituted formula under the grid, and the formula steps in the readout all update as you drag.
What to notice: The x-coordinate of M is the average of the two x-coordinates, and the y-coordinate of M is the average of the two y-coordinates — the two averages are separate, and the answer is an ordered pair because the midpoint is a point.
What to do: Each problem gives you a figure and a question about a midpoint or a missing endpoint. Work in three steps: (1) choose the equation to use to solve the problem, (2) find the coordinate the step asks for, and (3) report the point the question actually asks for. The gold highlight in the diagram marks the part the current step works on, and a value found in an earlier step is written at the top of the diagram. Each choice you pick moves a marker in the diagram to the point that choice describes, whether the choice is right or wrong, so you can see what the choice means; a missed choice is shown as a dashed ghost marker. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all three steps are complete, the full worked solution appears. Press New Item for the next problem: the four problems take turns in order, and the problem number is shown with the step question.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Midpoint Formula — Mr. J Geometry. All activities on this page run entirely in your browser.