CBSE Class 8 Mathematics Coordinate Geometry Notes Set 06

Cartesian Plane

Two number lines, the horizontal X-axis and the vertical Y-axis, cross at their zeros. The crossing point is the origin O(0, 0). The axes divide the plane into four quadrants numbered anticlockwise: I (+, +), II (−, +), III (−, −) and IV (+, −).
A point is written as (x, y). The x-coordinate (abscissa) is the distance from the Y-axis, measured along the X-axis, and the y-coordinate (ordinate) is the distance from the X-axis. Points on the X-axis have the form (x, 0) and points on the Y-axis have the form (0, y).

Plotting a Point

To plot (a, b) start at the origin, move a units along the X-axis (right if a is positive, left if negative) and then b units parallel to the Y-axis (up if b is positive, down if negative). The distance of (a, b) from the Y-axis is |a| and from the X-axis is |b|.

Distance Formula

The distance between P(x₁, y₁) and Q(x₂, y₂) is PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from the Pythagoras theorem applied to the right triangle formed by horizontal and vertical lines through P and Q.
The distance of P(x, y) from the origin is √(x² + y²). The distance is never negative, so only the positive root is taken.

Section Formula and Mid-point

The point dividing the segment from (x₁, y₁) to (x₂, y₂) internally in the ratio m : n is ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)).
For m = n = 1 this gives the mid-point ((x₁ + x₂)/2, (y₁ + y₂)/2). The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

Area of a Triangle and Collinearity

The area of the triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Three points are collinear exactly when this area is zero, or when one of the three distances equals the sum of the other two.

Recognising Figures from Vertices

Isosceles triangle: two equal sides. Equilateral triangle: three equal sides. Right triangle: the squares of two sides add up to the square of the third.
Parallelogram: opposite sides equal (or the diagonals have the same mid-point). Rectangle: a parallelogram with equal diagonals. Rhombus: all four sides equal. Square: four equal sides and equal diagonals.

Worked Examples

Example 1: Find the point on the x-axis equidistant from A(2, −5) and B(−2, 9). Let the point be (x, 0). Then (x − 2)² + 25 = (x + 2)² + 81, so −8x = 56 and x = −7. The point is (−7, 0).
Example 2: Show that (1, −1), (5, 2) and (9, 5) are collinear. The distances are AB = 5, BC = 5 and AC = 10, and AC = AB + BC, so the points lie on one line.
Example 3: If the distance between (x, 3) and (5, 7) is 5, then (x − 5)² + 16 = 25, so x − 5 = ±3 and x = 2 or 8.
Example 4: The vertices of a triangle are (a, a), (a + 1, a + 1) and (a + 2, a). Its area is ½ |a(1) + (a + 1)(0) + (a + 2)(−1)| = ½ × 2 = 1 square unit, whatever the value of a.