Coordinate Geometry
Coordinate Geometry
Coordinate geometry applies algebraic methods to solve geometric problems on axes. The key tools are: the distance formula, the midpoint formula, the gradient formula and the equation of a straight line. These connect the abstract properties of shapes to specific numerical values that can be calculated and verified.
Key Formulae
| Formula | Expression |
|---|---|
| Distance between \( (x_1,y_1) \) and \( (x_2,y_2) \) | \( d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \) |
| Midpoint of \( (x_1,y_1) \) and \( (x_2,y_2) \) | \( M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \) |
| Gradient of line through two points | \( m=\frac{y_2-y_1}{x_2-x_1} \) |
| Equation of a line | \( y=mx+c \) or \( y-y_1=m(x-x_1) \) |
| Perpendicular gradient | \( m_2=-\frac{1}{m_1} \) (product of gradients = \( -1 \)) |
Geometric Applications
- Proving a shape is a specific type (e.g. a square): show all sides equal (distance formula) and all angles 90° (perpendicular gradients).
- Finding the equation of a perpendicular bisector: find midpoint and perpendicular gradient; substitute into \( y-y_1=m(x-x_1) \).
- Proving points are collinear: show the gradient between each consecutive pair is the same.
Worked Examples
Find the distance between \( A(1,2) \) and \( B(7,10) \). Find the midpoint.
\[ AB=\sqrt{(7-1)^2+(10-2)^2}=\sqrt{36+64}=\sqrt{100}=10 \] \[ M=\left(\frac{1+7}{2},\frac{2+10}{2}\right)=(4,6) \]Find the equation of the line through \( (3,1) \) and \( (7,9) \). Find the equation of the perpendicular through \( (3,1) \).
\[ m=\frac{9-1}{7-3}=2 \implies y-1=2(x-3) \implies y=2x-5 \] \[ m_{\perp}=-\tfrac{1}{2} \implies y-1=-\tfrac{1}{2}(x-3) \implies y=-\tfrac{1}{2}x+\tfrac{5}{2} \]Show that \( A(0,0) \), \( B(4,0) \), \( C(4,4) \), \( D(0,4) \) is a square.
All sides: \( AB=BC=CD=DA=4 \) (distance formula). \( AC \) and \( BD \) diagonals equal: \( \sqrt{32}=4\sqrt{2} \). Gradients: \( AB \) gradient 0; \( BC \) gradient undefined (vertical) → perpendicular. All angles 90°. ✓ Square.
Key Takeaways
- Distance: \( \sqrt{(\Delta x)^2+(\Delta y)^2} \). Midpoint: average of each coordinate.
- Gradient: \( m=\Delta y/\Delta x \). Perpendicular gradient: \( -1/m \).
- Line through one point: \( y-y_1=m(x-x_1) \). Re-arrange to \( y=mx+c \) if needed.
- Perpendicular bisector: midpoint + perpendicular gradient.
- To prove a quadrilateral: calculate all side lengths and gradients; match to the property set of the target shape.