Important Formulas: Coordinate Geometry | Mathematics (Maths) Class 9 PDF Download

Introduction

Coordinate geometry plays a vital role in mathematics by visually representing geometric shapes in a two-dimensional plane and enhancing understanding of their inherent properties. In this exploration, we delve into the coordinate plane and the concept of point coordinates, aiming to provide an introductory understanding of the realm of coordinate geometry.

Coordinate Plane

The Cartesian plane, or coordinate plane, divides the plane into two dimensions using the horizontal x-axis and the vertical y-axis. This system allows precise location identification, with the intersection point termed the origin (0, 0). Points on the coordinate plane are represented as (x, y), where 'x' denotes the position relative to the x-axis, and 'y' indicates the position concerning the y-axis. The plane is further divided into four quadrants by these axes.

Distance Formula

The distance (D) between two points (x1, y1) and (x2, y2) is computed as the square root of the sum of squared differences in their x and y coordinates:

This formula, derived from Pythagoras' theorem, expresses the distance between two points in coordinate geometry.

Slope Formula

The slope of a line can be calculated using the angle it forms with the positive x-axis or by selecting two points on the line.

Slope (m) = Tanθ = (y2 – y1) / (x2 – x1)

Midpoint Formula

This formula determines the midpoint of a line joining two points (x1, y1) and (x2, y2). The midpoint’s coordinates are the averages of the x and y coordinates of the given points.

Given two points A (x1, y1) and B (x2, y2), the midpoint between A and B is given by,

M(x3, y3) = ((x1 + x2)/2, (y1 + y2)/2)

where M is the midpoint between A and B, and (x3, y3) are its coordinates.

Midpoint (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2) 

Section Formula

The section formula identifies the coordinates of a point that divides the line segment between two points (x1, y1) and (x2, y2) in a given ratio m:n

Point (x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))

Centroid Formula

The centroid of a triangle formed by vertices A (x1, y1), B (x2, y2), and C (x3, y3) can be found using the formula:

Centroid of a Triangle (x, y) = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

Coordinate the Geometry Area of a Triangle Formula

Area of a Triangle Formula: The area of a triangle with vertices A (x1, y1), B (x2, y2), and C (x3, y3) is computed using the formula:

Area = 0.5 * |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

1. We require two perpendicular axes to locate a point in the plane. One of them is horizontal and other is Vertical
2. The plane is called Cartesian plane and axis are called the coordinates axis
3. The horizontal axis is called x-axis and Vertical axis is called Y-axis
4. The point of intersection of axis is called origin.
5. The distance of a point from y axis is called x –coordinate or abscissa and the distance of the point from x –axis is called y – coordinate or Ordinate
6. The distance of a point from y axis is called x –coordinate or abscissa and the distance of the point from x –axis is called y – coordinate or Ordinate
7. The Origin has zero distance from both x-axis and y-axis so that its abscissa and ordinate both are zero. So the coordinate of the origin is (0, 0)
8. A point on the x –axis has zero distance from x-axis so coordinate of any point on the x-axis will be (x, 0)
9. A point on the y –axis has zero distance from y-axis so coordinate of any point on the y-axis will be (0, y)
10. The axes divide the Cartesian plane in to four parts. These Four parts are called the quadrants 

The coordinates of the points in the four quadrants will have sign according to the below table