5. ANALYTICAL GEOMETRY: POINTS

Introduction to Analytical Geometry

Analytical geometry (also called coordinate geometry) studies geometric figures using the coordinate system. A point is represented using (x, y) in a plane. Every location in the plane can be described using coordinates.

Cartesian Coordinate System

  • The plane is divided by two perpendicular axes: X-axis (horizontal) and Y-axis (vertical).
  • The point where axes intersect is called the origin (0, 0).
  • Any point P is written as P(x, y), where:
    • x = horizontal distance from origin
    • y = vertical distance from origin

Quadrants of the Coordinate Plane

The axes divide the plane into four quadrants:

  • 1st Quadrant: x > 0, y > 0
  • 2nd Quadrant: x < 0, y > 0
  • 3rd Quadrant: x < 0, y < 0
  • 4th Quadrant: x > 0, y < 0

Distance Between Two Points

For points P(x1, y1) and Q(x2, y2):

PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]

This is the Distance Formula.

Midpoint of a Line Segment

The midpoint M of points P(x1, y1) and Q(x2, y2) is:

M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Section Formula

If a point divides the line joining P(x1, y1) and Q(x2, y2) in the ratio m : n:

Internal Division

((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))

External Division

((mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n))

Centroid of a Triangle

For triangle with vertices:

A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)

The centroid G is given by:

G = ((x₁ + x₂ + x₃)/3 , (y₁ + y₂ + y₃)/3)

Collinearity of Points

Three points A, B, C are collinear if area of triangle ABC = 0.

Area formula:

Area = (1/2) | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

Section of Coordinate Axes

If a line meets the X-axis at A(a, 0) and Y-axis at B(0, b), it divides the plane and helps in writing equations of lines.

Slope of a Line:

Though detailed slope formulas appear in the next chapter, the this topic introduces the idea visually:

  • Slope represents the “steepness” of a line.
  • Slope is positive when the line rises from left to right.
  • Slope is negative when the line falls from left to right.

Examples:

This topic includes many examples involving:

  • Distance between two points
  • Midpoint calculation
  • Internal and external section formula
  • Centroid of triangle
  • Checking if points are collinear

Typical steps shown in examples:

  1. Write coordinates clearly.
  2. Select correct formula (distance/section/midpoint).
  3. Substitute values and simplify.
  4. Interpret result (length, ratio point, etc.).

Detailed Notes:

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