Distance Formula
Distance between two points P(x1, y1) and Q(x2, y2)
PQ = √[(x2 - x1)² + (y2 - y1)²]
💡 Remember:
This is just Pythagoras theorem! The horizontal distance is (x2-x1) and vertical distance is (y2-y1). The straight line is the hypotenuse.
- Distance from origin: If point is (x,y), distance from (0,0) = √(x² + y²)
- Collinearity Check: Three points A, B, C are collinear if AB + BC = AC (or any combination where sum of two equals third)
- Types of Triangles:
• Equilateral: All 3 sides equal
• Isosceles: Any 2 sides equal
• Scalene: All sides different
• Right-angled: a² + b² = c²
Section Formula
Point P dividing A(x1,y1) and B(x2,y2) internally in ratio m1:m2
P = [(m1x2 + m2x1)/(m1+m2) , (m1y2 + m2y1)/(m1+m2)]
💡 Remember:
The ratio m1:m2 means the point is m1 parts towards B and m2 parts towards A. If ratio is 2:3, point is closer to B.
- Midpoint Formula: When ratio is 1:1
M = [(x1+x2)/2 , (y1+y2)/2] - Centroid of Triangle: Intersection point of medians
G = [(x1+x2+x3)/3 , (y1+y2+y3)/3] - Trisection Points: Divide in ratios 1:2 and 2:1
How to Identify Quadrilaterals Using Distance Formula
▢ Square
- ✓ All 4 sides equal
- ✓ Both diagonals equal
- ✓ Diagonals perpendicular (optional check)
▭ Rectangle
- ✓ Opposite sides equal
- ✓ Both diagonals equal
- ✗ Adjacent sides NOT equal
◆ Rhombus
- ✓ All 4 sides equal
- ✗ Diagonals NOT equal
- ✓ Diagonals bisect each other at 90°
▱ Parallelogram
- ✓ Opposite sides equal
- ✓ Diagonals bisect each other
- ✗ Adjacent sides NOT equal
- ✗ Diagonals NOT equal
Exam Tips & Common Mistakes to Avoid
Always Draw Diagram
Sketch rough figure before solving. It helps visualize and avoid sign errors.
Watch Signs Carefully
(-3) - (-5) = +2, not -8. Double check negative number operations.
Simplify Radicals
√8 = 2√2, √32 = 4√2. Always simplify your final answer.
Collinearity Check
Use area method: Area = ½|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|. If 0, points are collinear.
Equidistant Points
Set PA² = PB² (not PA = PB) to avoid square roots. Much easier!
Point on X-axis
Any point on x-axis has y-coordinate = 0. So use (x, 0).