Coordinate Geometry

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Distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] This formula is derived from the Pythagorean theorem and gives the straight-line distance between any two points in a coordinate plane.

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Using distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] d = √[(7 - 3)² + (1 - 4)²] d = √[4² + (-3)²] d = √[16 + 9] d = √25 = 5 units

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For points A(x₁, 0) and B(x₂, 0) on the X-axis: Distance = |x₂ - x₁| We take the absolute value because distance is always positive. The y-coordinates are both 0, so we only consider the difference i

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For points A(0, y₁) and B(0, y₂) on the Y-axis: Distance = |y₂ - y₁| We take the absolute value because distance is always positive. The x-coordinates are both 0, so we only consider the difference i

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If point P divides line segment AB internally in the ratio m₁:m₂, then: P(x, y) = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) where A(x₁, y₁) and B(x₂, y₂) are the endpoints.

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

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Using section formula with m₁ = 1, m₂ = 3: x = (1×6 + 3×2)/(1+3) = (6+6)/4 = 3 y = (1×7 + 3×3)/(1+3) = (7+9)/4 = 4 Therefore, the point is P(3, 4).

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Trisection points are the two points that divide a line segment into three equal parts. If AB is divided into three equal parts by points P and Q: - P divides AB in ratio 1:2 - Q divides AB in ratio