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Coordinate Geometry
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Intersection of lines
Intersection (both y = mx + c)
\[ y = 2x + 1,\; y = 7 – x \]
Solve simultaneous equations (both y = mx + c).
Intersection (both ax + by = c)
\[ 2x + 3y = 12,\; x – y = 1 \]
Solve simultaneous equations (both ax + by = c).
Intersection with x-axis
\[ y = 6 – 2x \text{ meets } x\text{-axis} \] \[ 3x – 2y = 9 \text{ meets } x\text{-axis} \]
Find where a line crosses the x-axis (y = mx + c or ax + by = c).
Intersection with y-axis
\[ y = 5 – 3x \text{ meets } y\text{-axis} \] \[ 2x – 3y = 12 \text{ meets } y\text{-axis} \]
Find where a line crosses the y-axis (y = mx + c or ax + by = c).
Intersection with fractional coordinates
\[ y = 2x + 1,\; y = 5x – 2 \]
Find intersection giving exact fractions.
Points on lines
Does point lie on line?
\[ (3, 1) \text{ on } y = 7 – 2x \text{?} \] \[ (2, 3) \text{ on } 3x – y = 5 \text{?} \]
Check if coordinates satisfy a line’s equation (y = mx + c or ax + by = c).
Find k if (k, …) lies on line
\[ (k, 5) \text{ on } y = 8 – 3x \]
Find x-coordinate given y-coordinate.
Find k if (…, k) lies on line
\[ (4, k) \text{ on } 2x + 3y = 14 \]
Find y-coordinate given x-coordinate.
Find integer points on a line
\[ y = 1 – \frac{1}{2}x,\; -4 \leq x \leq 6 \]
List all integer coordinate points in range.
Collinearity
Collinear using gradient
\[ A(1,2), B(3,6), C(5,10) \]
Compare gradients to check collinearity.
Collinear using substitution
\[ A(2,5), B(4,9), C(7,15) \]
Find line AB, then check if C lies on it.
Geometric applications
Fourth vertex of parallelogram
\[ A(1,2), B(4,3), C(6,7) \to D \]
Use equal gradients to find D.
Fourth vertex of rectangle
\[ A(0,0), B(4,2), C(3,6) \to D \]
Use perpendicular gradients to find D.
Perpendicular bisector
\[ A(2,3), B(6,7) \]
Find equation through midpoint, perpendicular.
Is triangle right-angled?
\[ A(1,1), B(4,5), C(8,2) \]
Check for perpendicular gradients.
Timer (Optional)
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