Co-ordinate Geometry Cheatsheet

Cheat Codes & Shortcuts

• Distance: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
• Midpoint: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
• Slope (Gradient): \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Equation of Line (2-point): \( y - y_1 = m(x - x_1) \)
• General Form of Line: \( Ax + By + C = 0 \)
• Circle (centre \( (h,k) \), radius \( r \)): \( (x-h)^2 + (y-k)^2 = r^2 \)
• PQ Formula (perpendicular distance to line): \( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2+B^2}} \)
• Area of Triangle: \( \text{Area} = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \)
• Locus: Geometric place of points satisfying a condition.
• Collinearity Check: Points are collinear if area from above formula is 0.

Quick Reference Table

Concept	Formula / Equation	Example / Use
Distance	\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)	Between \( (1,2) \), \( (4,6) \): \( d = 5 \)
Midpoint	\( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)	\( (1,2), (3,4) \rightarrow (2,3) \)
Slope	\( m = \frac{y_2 - y_1}{x_2 - x_1} \)	\( (2,3), (4,7) \rightarrow m = 2 \)
Line Eq.	\( y - y_1 = m(x - x_1) \)	Through \( (1,2), m=3 \): \( y - 2 = 3(x - 1) \)
Circle	\( (x-3)^2 + (y+1)^2 = 9 \)	Centre: \( (3, -1) \), Radius: \( 3 \)
Perpendicular Distance	\( d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}} \)	From \( (2,2) \) to \( x+y=4 \): \( d=\sqrt{2} \)

Advice

Co-ordinate Geometry Quick Tips

• Line between points: Find slope, use \( y-y_1=m(x-x_1) \)
• Circle from points: Use distance formula for radius.
• Collinearity: Area formula zero means points are collinear.
• Perpendicular/Parallel: Slopes \( m_1 \), \( m_2 \); perpendicular iff \( m_1 \cdot m_2 = -1 \), parallel iff \( m_1 = m_2 \)
• Midpoint and Section Formula: For ratios, use weighted averages for coordinates.

Co-ordinate Geometry Speed Quiz

Test your speed with 5 co-ordinate geometry questions! You have 30 seconds per question.