Rectangular Cartesian Coordinates
1. Straight-line toolkit
Most coordinate questions are chains of gradient → equation → intersection → distance/area.
Gradient
\[m=\frac{y_2-y_1}{x_2-x_1}\]Point-gradient form
\[y-y_1=m(x-x_1)\]Parallel
\[m_1=m_2\]Perpendicular
\[m_1m_2=-1\quad(\text{non-vertical lines})\]Equation of a perpendicular line
A line through \((2,-1)\) is perpendicular to \(3x-2y+7=0\). The given line has gradient \(3/2\), so the new gradient is \(-2/3\): \[y+1=-\frac23(x-2).\]Common mistake. A vertical line has undefined gradient and equation \(x=c\). Do not force it into \(y=mx+c\).
2. Distance, midpoint and section formula
Coordinate section formula is the component version of the vector section formula.
Distance
\[AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]Midpoint
\[M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\]Internal division
\[AP:PB=m:n\Rightarrow P=\left(\frac{nx_A+mx_B}{m+n},\frac{ny_A+my_B}{m+n}\right)\]Ratio point
If \(A=(1,4),B=(11,-1)\) and \(AP:PB=2:3\), then \[P=\left(\frac{3(1)+2(11)}5,\frac{3(4)+2(-1)}5\right)=(5,2).\]3. Coordinate areas and geometric conditions
Coordinate geometry rewards a clean sequence of small exact calculations.
Triangle from coordinates
\[\text{Area}=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|\]- Find any missing coordinates from line equations or ratios.
- Use gradients to prove parallel/perpendicular conditions.
- Use distance for equal-length conditions.
- For an area, choose a simple base-height route if available; otherwise use the coordinate determinant formula.
Exam tip. Keep exact fractions and surds through the coordinate work. Early rounding can destroy a later exact gradient or area.
4. Equidistance and perpendicular bisectors
Distance conditions can be turned directly into line equations.
Equidistant point condition
If \(P(x,y)\) is equidistant from \(A\) and \(B\), then \(PA^2=PB^2\). Squaring removes the square roots and simplifies to the equation of the perpendicular bisector of \(AB\).Exam tip. Use squared distances when comparing lengths. It avoids unnecessary radicals and is algebraically equivalent for nonnegative distances.