Distance and Midpoint Formula

Coordinate Geometry Formulas

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The distance and midpoint formulas help you analyze line segments on the coordinate plane. They let you find how far apart two points are and the exact point halfway between them. These ideas are essential in geometry, algebra, graphing, and many real world applications such as mapping and design. Both formulas come directly from the coordinates of the endpoints.

The distance formula is based on the Pythagorean theorem, using the horizontal and vertical changes between two points as the legs of a right triangle. The midpoint formula works by averaging the x-coordinates and averaging the y-coordinates. When you look at a segment on a graph, the same two endpoints can tell you both its length and its center. This makes coordinate geometry a powerful way to connect algebra and visual reasoning.

Key Facts

Distance between $A(x_1, y_1)$ and $B(x_2, y_2)$ : $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint of $A(x_1, y_1)$ and $B(x_2, y_2)$ : $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Horizontal change is $\Delta x = x_2 - x_1$
Vertical change is $\Delta y = y_2 - y_1$
Distance formula comes from $d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$
The midpoint is found by averaging each coordinate separately

Vocabulary

Coordinate plane: A flat grid formed by the x-axis and y-axis where points are located using ordered pairs.
Ordered pair: A pair of numbers written as (x, y) that gives the location of a point on the plane.
Distance formula: A formula that gives the length of the segment between two points on the coordinate plane.
Midpoint: The point exactly halfway between the endpoints of a segment.
Pythagorean theorem: A theorem stating that for a right triangle, $a^2 + b^2 = c^2$ , which is used to derive the distance formula.

Common Mistakes to Avoid

Mixing x-coordinates with y-coordinates, which is wrong because x-values must be compared with x-values and y-values with y-values only.
Forgetting to square both coordinate differences in the distance formula, which is wrong because the formula depends on $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$ before adding.
Adding the coordinates and forgetting to divide by 2 for the midpoint, which is wrong because the midpoint is the average of each coordinate.
Dropping negative signs when subtracting coordinates, which is wrong because a mistake in signs changes both the distance and the midpoint.

Practice Questions

1 Find the distance between the points (2, 3) and (8, 11).
2 Find the midpoint of the segment with endpoints (-4, 6) and (10, -2).
3 Two points have the same y-coordinate but different x-coordinates. Explain what the segment looks like and how the distance and midpoint can be found more simply.