Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows us to represent geometric shapes algebraically and solve geometric problems using algebraic methods.

1. Midpoint and Gradient of a Line Segment

The Midpoint: The midpoint is the point that divides a line segment into two equal halves. For two points P(x₁, y₁) and Q(x₂, y₂), the coordinates of the midpoint M are calculated as:

• M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

The Gradient (Slope): The gradient (m) represents the steepness of a line. It is the ratio of the vertical change (rise) to the horizontal change (run). For points P(x₁, y₁) and Q(x₂, y₂):

• m = (y₂ - y₁) / (x₂ - x₁)

2. Distance Between Two Points

The distance (d) between two points P(x₁, y₁) and Q(x₂, y₂) is derived from the Pythagorean theorem and is given by:

• d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: Find the distance between A(2, 3) and B(5, 7).
d = √[(5 - 2)² + (7 - 3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units.

3. Parallel and Perpendicular Lines

The relationship between the gradients of two lines determine their geometric relationship:

• Parallel Lines: Two lines are parallel if their gradients are equal (m₁ = m₂).
• Perpendicular Lines: Two lines are perpendicular if the product of their gradients is -1 (m₁ × m₂ = -1). This means the gradient of one is the negative reciprocal of the other (m₁ = -1/m₂).

4. Equations of Straight Lines

There are several ways to express the equation of a straight line depending on the information given:

• Slope-Intercept Form: y = mx + c (where m is the gradient and c is the y-intercept).
• Point-Slope Form: y - y₁ = m(x - x₁) (used when you have one point and the gradient).
• Two-Point Form: (y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁) (used when you have two points).
• General Form: Ax + By + C = 0 (where A, B, and C are constants).

Example: Find the equation of the line passing through (2, 4) with a gradient of 3.
Using y - y₁ = m(x - x₁):
y - 4 = 3(x - 2)
y - 4 = 3x - 6
y = 3x - 2.