Midpoint Formula

The midpoint of a line segment is the point exactly halfway between its two endpoints. In coordinate geometry, we use the midpoint formula to find this point quickly and accurately.

It is useful for dividing a segment into equal halves, finding centers of shapes, and solving geometric problems.

Suppose a line segment has endpoints:

\(A(x_1, y_1)\) and \(B(x_2, y_2)\)

The midpoint M is the point that is the average of the x-coordinates and the y-coordinates.

The coordinates of the midpoint \(M(x, y)\) of a segment joining:

\(A(x_1, y_1)\) and \(B(x_2, y_2)\)

are given by:

\(M = \left( \dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2} \right) \)

Just add the x-values and divide by 2, and do the same for the y-values.

Let’s understand the formula with easy examples.

The midpoint is the center point of the segment AB.

• It divides the segment into two equal lengths.
• It lies exactly halfway, no matter the direction or quadrant.
• It can represent the center of diagonals in rectangles, parallelograms, and squares.

• Adding instead of averaging (forgetting to divide by 2).
• Mixing up x and y coordinates.
• Applying midpoint formula when the question asks for section formula or vice versa.
• Arithmetic mistakes with negative numbers.

Find the midpoint of the following:

1. A(1, 3) and B(9, 11)
2. A(-5, 4) and B(3, -8)
3. A(0, 0) and B(6, -4)
4. A(7, -2) and B(7, 10)

• Midpoint is the point exactly halfway between two points.
• Formula: \(M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)\)
• Midpoint represents average of x and y coordinates.
• Useful for geometry problems and understanding symmetry.