Find the Midpoint of a Segment: Step-by-Step Guide
The midpoint of a line segment is the point exactly halfway between its two endpoints. To find it, take the average of the x-coordinates and the average of the y-coordinates: M = ((x1 + x2)/2, (y1 + y2)/2).
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Label the endpoints. Call the two endpoints (x1, y1) and (x2, y2).\[(x_1, y_1),\;(x_2, y_2)\]
- Average the x-coordinates. Add x1 and x2, then divide by 2. This is the x-coordinate of the midpoint.\[x_M = \dfrac{x_1 + x_2}{2}\]
- Average the y-coordinates. Add y1 and y2, then divide by 2. This is the y-coordinate of the midpoint.\[y_M = \dfrac{y_1 + y_2}{2}\]
- Write the midpoint. Put the two answers together as a point: (x, y).\[M = \left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right)\]
Worked example
Find the midpoint of the segment with endpoints (-2, 5) and (4, -1).
Step 1: Call the endpoints (x1, y1) = (-2, 5) and (x2, y2) = (4, -1).
Step 2: The x-coordinate of the midpoint: (-2 + 4) / 2 = 2 / 2 = 1.
Step 3: The y-coordinate of the midpoint: (5 + (-1)) / 2 = 4 / 2 = 2.
Step 4: The midpoint is (1, 2).
Check: The distance from (-2, 5) to (1, 2) is the square root of (9 + 9), and the distance from (1, 2) to (4, -1) is also the square root of (9 + 9). They are equal, so (1, 2) is exactly in the middle.
Tips and common mistakes
- ADD the coordinates and divide by 2. Do not subtract. Subtracting is for slope and distance.
- Be careful with negative numbers when you add.
- The midpoint is a point with two numbers, (x, y), not a single number.
- If you know the midpoint M and one endpoint A, you can find the other endpoint: double M's coordinates, then subtract A's coordinates.
- The same idea works on a number line. The midpoint of a and b is (a + b) / 2.
- To check, make sure the midpoint is the same distance from both endpoints.