Coordinate Geometry
This section covers Coordinate Geometry. The exam tests the distance formula, section formula, midpoint formula, and area of a triangle from coordinates.
Core formulas
For points A(x₁,y₁) and B(x₂,y₂):
- Distance AB = √[(x₂ - x₁)² + (y₂ - y₁)²].
- Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Section formula for dividing AB in ratio m:n internally: ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). For triangle vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.
Worked example
Find the distance between A(1,2) and B(4,6). Difference in x is 3 and in y is 4, so AB = √(3² + 4²) = √25 = 5. The midpoint is ((1+4)/2, (2+6)/2) = (2.5, 4). For triangle O(0,0), P(4,0), Q(0,3), area = ½|0(0-3) + 4(3-0) + 0(0-0)| = ½×12 = 6 square units.
Traps
Subtract coordinates in the same order; reversing one pair changes signs. The section formula uses the ratio in the order “from A to B”; swapping m and n gives a different point. Area must be positive, so keep the absolute value. Do not confuse the midpoint with a point dividing a segment in ratio 1:1-they agree, but the section formula must still be applied consistently.