Coordinate geometry foundations

Coordinate geometry connects algebra and geometry by representing points with ordered pairs \((x,y)\) on the Cartesian plane.

The Cartesian plane

Horizontal axis is \(x\)-axis, vertical axis is \(y\)-axis, and intersection is origin \((0,0)\).

Quadrants:

• Q1: \((+,+)\)
• Q2: \((-,+)\)
• Q3: \((-,-)\)
• Q4: \((+,-)\)

Plotting and reading points

Ordered pair \((a,b)\) means move \(a\) units along \(x\), then \(b\) units along \(y\).

Example: \((-3,2)\) lies in Quadrant II.

Distance between two points

For \(A(x_1,y_1)\), \(B(x_2,y_2)\):

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

This comes from Pythagoras' theorem on horizontal and vertical differences.

Midpoint of a segment

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

Example: Midpoint of \((2,5)\) and \((8,-1)\) is \((5,2)\).

Gradient (slope) of a line

For points \(A(x_1,y_1)\), \(B(x_2,y_2)\) with \(x_2\ne x_1\):

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

• \(m>0\): line rises left to right.
• \(m<0\): line falls left to right.
• \(m=0\): horizontal line.
• Vertical line has undefined slope.

Equations of lines

Slope-intercept form:

\[ y=mx+c \]

where \(m\) is slope and \(c\) is \(y\)-intercept.

Point-slope form:

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

Parallel and perpendicular lines

• Parallel lines have equal slopes: \(m_1=m_2\).
• Perpendicular lines satisfy \(m_1m_2=-1\) (when both slopes are defined).

Why these formulas work (derivation)

Distance formula from Pythagoras

Between \(A(x_1,y_1)\) and \(B(x_2,y_2)\), horizontal change is \(\Delta x=x_2-x_1\) and vertical change is \(\Delta y=y_2-y_1\). These are the legs of a right triangle, and segment \(AB\) is its hypotenuse.

\[ AB^2=(\Delta x)^2+(\Delta y)^2 \]

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

Midpoint as averaging coordinates

A midpoint lies exactly halfway in both horizontal and vertical directions, so each coordinate is an average of the endpoints:

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

Slope formula from rise over run

Slope measures steepness: rise divided by run. For two points, rise is \(y_2-y_1\) and run is \(x_2-x_1\), giving:

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

This is why vertical lines (\(x_2=x_1\)) have undefined slope: the denominator becomes zero.

Exam strategy

• Write coordinates clearly before substituting formulas.
• Keep exact form (roots/fractions) unless decimal is requested.
• Check whether slope is undefined for vertical lines.
• Use diagrams to avoid sign mistakes in differences.

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

• Maps/GPS: locations represented using coordinate systems.
• Computer graphics: points and lines placed on pixel coordinate planes.
• Physics: trajectories and motion graphs depend on coordinates and slope.

References

• Standard high-school coordinate geometry units.
• Analytic geometry references on points, lines, distance, and midpoint.