Distance and midpoint formulas
Distance and Midpoint Formulas
Distance and Midpoint Formulas is the opening topic of Coordinate Geometry in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.2(B), which requires students to derive and use the distance and midpoint formulas to verify geometric relationships.
The distance formula \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) gives a segment's length, and the midpoint formula averages the coordinates.
Theory
For points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
- Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) — the Pythagorean theorem applied to the horizontal and vertical changes.
- Midpoint: \(M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\) — the average of the coordinates.
The two formulas:
How to use the formulas
- Label the points \((x_1,y_1)\) and \((x_2,y_2)\).
- For distance, subtract, square, add, and take the square root.
- For the midpoint, average each coordinate.
- To find an endpoint, use \(x_B=2x_M-x_A\) (and the same for \(y\)).
Use the distance formula.
| \(d\) | \(=\) | \(\sqrt{(4-1)^2+(5-1)^2}\) |
| \(=\) | \(\sqrt{3^2+4^2}\) | |
| \(=\) | \(\sqrt{9+16}=\sqrt{25}=5\) |
Average the coordinates.
| \(M\) | \(=\) | \(\left(\dfrac{1+4}{2},\ \dfrac{1+5}{2}\right)\) |
| \(=\) | \(\left(\dfrac{5}{2},\ 3\right)\) |
Substitute carefully with the signs.
| \(d\) | \(=\) | \(\sqrt{(4-(-2))^2+(-5-3)^2}\) |
| \(=\) | \(\sqrt{6^2+(-8)^2}\) | |
| \(=\) | \(\sqrt{36+64}=\sqrt{100}=10\) |
Each midpoint coordinate is the average, so double it and subtract \(A\).
| \(x_B\) | \(=\) | \(2(3)-1=5\) |
| \(y_B\) | \(=\) | \(2(4)-2=6\) |
Common pitfalls
Frequently asked questions
What is the distance formula?
\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), the Pythagorean theorem on the coordinate changes.
What is the midpoint formula?
\(M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\): the average of the coordinates.
How do you find a missing endpoint from the midpoint?
Double each midpoint coordinate and subtract the known endpoint: \(x_B=2x_M-x_A\).
Why does the distance formula work?
The horizontal and vertical changes are the legs of a right triangle, so the distance is the hypotenuse.