Co-ordinate Geometry

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Distance formula: d(A,B) = √[(x₂ - x₁)² + (y₂ - y₁)²] This formula is derived using Pythagoras theorem. It works for any two points in the coordinate plane, regardless of their position relative to a

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Using distance formula: d(P,Q) = √[(7-3)² + (1-4)²] = √[4² + (-3)²] = √[16 + 9] = √25 = 5 units The distance between P(3, 4) and Q(7, 1) is 5 units.

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Distance from origin: d(O,P) = √[(a-0)² + (b-0)²] = √[a² + b²] This is a special case of the distance formula where one point is the origin (0, 0).

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Section Formula (Internal Division): If point P divides line segment AB internally in ratio m:n, then: P(x,y) = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)) where A(x₁, y₁) and B(x₂, y₂) are the endpoints.

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Using section formula with m = 2, n = 1: P(x,y) = ((2×8 + 1×2)/(2+1), (2×9 + 1×3)/(2+1)) = ((16 + 2)/3, (18 + 3)/3) = (18/3, 21/3) = (6, 7) Point P has coordinates (6, 7).

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Midpoint Formula: For line segment joining points A(x₁, y₁) and B(x₂, y₂), the midpoint M is: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) This is a special case of section formula when m = n = 1 (ratio 1:1).

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Using midpoint formula: M = ((-3 + 5)/2, (2 + (-4))/2) = (2/2, -2/2) = (1, -1) The midpoint is M(1, -1).

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Centroid Formula: For triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the centroid G is: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) The centroid divides each median in the ratio 2:1.