Cartesian System of Coordinates

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The Cartesian System of Coordinates is a coordinate system that uses two perpendicular lines (axes) to locate points in a plane. It is named after French mathematician René Descartes. The system consi

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Quadrant sign conventions: • I Quadrant: x > 0, y > 0 (both positive) • II Quadrant: x < 0, y > 0 (negative x, positive y) • III Quadrant: x < 0, y < 0 (both negative) • IV Quadrant: x > 0, y < 0 (pos

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Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²] Step-by-step solution: Given: A(3,4) and B(7,1) Step 1: Identify coordinates x₁ = 3, y₁ = 4, x₂ = 7, y₂ = 1 Step 2: Apply formula d = √[(7-3)² + (1-4)²]

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Three points A, B, C are collinear if AB + BC = AC (or any similar combination) Method: Step 1: Calculate all three distances AB, BC, AC Step 2: Check if sum of any two equals the third Example: A(1

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Internal Division Formula: P(x,y) = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) Step-by-step solution: Given: A(2,3), B(8,7), ratio m₁:m₂ = 2:1 Step 1: Identify values x₁ = 2, y₁ = 3, x₂ = 8,

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Internal Division (point lies between A and B): P = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) External Division (point lies outside segment AB): P = ((m₁x₂ - m₂x₁)/(m₁ - m₂), (m₁y₂ - m₂y₁)/(

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Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2) (Special case of section formula with ratio 1:1) Step-by-step solution: Given: A(-2,5) and B(4,-1) Step 1: Identify coordinates x₁ = -2, y₁ = 5, x₂ = 4,

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Area Formula: Area = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| Step-by-step solution: Given: A(1,2), B(3,4), C(5,1) Step 1: Identify coordinates x₁=1, y₁=2, x₂=3, y₂=4, x₃=5, y₃=1 Step 2: Apply form