1. What is a Midpoint?
The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. It divides the segment into two equal parts.
In coordinate geometry, we use the midpoint formula to find this point when the coordinates of the endpoints are known.
1.1. Idea Behind the Midpoint
The midpoint simply averages the x-coordinates and the y-coordinates of the endpoints. This gives the center point of the line segment.
2. Deriving the Midpoint Formula
Consider a line segment with endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\). The midpoint lies exactly in the middle, so we take the average of the x-values and the y-values.
2.1. Averaging Coordinates
The midpoint has:
- x-coordinate = average of \(x_1\) and \(x_2\)
- y-coordinate = average of \(y_1\) and \(y_2\)
2.1.1. Visual Understanding
If you imagine sliding from \(x_1\) to \(x_2\), the midpoint is right at the halfway distance. Same idea applies to the y-values.
3. Midpoint Formula
The midpoint \(M\) of the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\( M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \)
3.1. Key Observations
- The midpoint always lies between the two points.
- If the endpoints have integer coordinates, the midpoint may or may not be an integer point.
- The formula works for all quadrants.
4. Examples
These examples illustrate how to apply the midpoint formula in different cases.
4.1. Example 1: Simple Coordinates
Find the midpoint of \((2, 4)\) and \((6, 10)\).
\( M = \left( \dfrac{2 + 6}{2}, \dfrac{4 + 10}{2} \right) = (4, 7) \)
4.2. Example 2: Negative Coordinates
Find the midpoint of \((-3, 5)\) and \((7, -1)\).
\( M = \left( \dfrac{-3 + 7}{2}, \dfrac{5 + (-1)}{2} \right) = (2, 2) \)
4.3. Example 3: Fractional Result
Find the midpoint of \((1, 2)\) and \((4, 3)\).
The midpoint is:
\( M = \left( \dfrac{1 + 4}{2}, \dfrac{2 + 3}{2} \right) = (2.5, 2.5) \)