History

Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected algebraic equations to geometric curves, creating a major foundation for modern mathematics.

Derivation and reasoning

Distance formula (step by step)

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

  1. Construct a right triangle by drawing horizontal and vertical guide lines.
  2. Its horizontal leg is \(|x_2-x_1|\), and vertical leg is \(|y_2-y_1|\).
  3. By Pythagoras, \(AB^2=(x_2-x_1)^2+(y_2-y_1)^2\).
  4. Take the positive square root because distance is nonnegative.

\[ AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

Midpoint formula (step by step)

A midpoint splits a segment into two equal parts in both directions. So the \(x\)-coordinate is halfway between \(x_1\) and \(x_2\), and the \(y\)-coordinate is halfway between \(y_1\) and \(y_2\).

\[ M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \]

Slope formula and line forms

Slope is rise over run. For points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):

\[ m=\frac{y_2-y_1}{x_2-x_1} \]

Rearranging the same idea for a generic point \((x,y)\) on the line through \((x_1,y_1)\) gives point-slope form:

\[ y-y_1=m(x-x_1) \]

Expanding gives slope-intercept form \(y=mx+c\), where \(c\) is the \(y\)-intercept.

Checkpoints

State the quadrant of \((-4,-7)\).

Answer: Quadrant III.

Find distance between \((1,2)\) and \((7,10)\).

Answer: \(\sqrt{6^2+8^2}=\sqrt{100}=10\).

Find midpoint of \((-3,5)\) and \((9,-1)\).

Answer: \(\left(3,2\right)\).

Find slope through \((2,-1)\) and \((6,7)\).

Answer: \(m=\frac{7-(-1)}{6-2}=\frac84=2\).

Line with slope \(3\) through \((1,4)\): find equation.

Answer: \(y-4=3(x-1)\Rightarrow y=3x+1\).

Explain in one sentence why the distance formula has a square root.

Answer: Pythagoras gives \(AB^2\), so we take the positive square root to get the actual distance \(AB\).

Why is slope undefined for a vertical line?

Answer: For a vertical line, \(\Delta x=0\), so \(m=\frac{\Delta y}{\Delta x}\) would require division by zero.

Applications

References