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Coordinate Geometry Formulas, Meaning and Examples

 

Coordinate Geometry is thought to be one of the most fascinating topics of mathematics and science. Coordinate geometry is referred to as analytical geometry, and it is applied to form connections between algebraic maths and geometry. These relations are generally conceived in the form of graphs and curves. Each type of algebraic equation has unique graphs. The ordered pair of numbers are used to demonstrate the coordinates of a point on the graph paper.

Introduction

Coordinate Points: These are points on the cartesian plane which define the exact location of a point. It is like an address of a point. For a 2-dimensional point, the coordinates are (x, y) and for a three-dimensional point, the coordinates are (x, y, z).

Coordinate Plane: For a two-dimensional point (x, y) the distance between ‘x’ and the origin is known as the abscissa, and the distance between ‘y’ and the origin is known as ordinate. The plane is divided into two dimensions by a Cartesian plane and is useful for locating points. The horizontal x-axis and vertical y-axis are the two axes of the coordinate system. These axes split the plane into four quadrants, and the intersection point is the origin of this axis (0, 0).

Formulas Related to Coordinate Geometry

There are primarily five formulas related to coordinate geometry that are explained below:

Coordinate Geometry Distance Formula

Let us consider two points on the coordinate plane A and B. Let the coordinates of point A be (a1, a2) and B be (b1, b2). According to the distance formula, the distance between 2 points is the square root of the sum of the square of the difference between the like terms. The x terms are subtracted together, and y terms are subtracted together and then squared.

D = √(a2−a1)2+(b2−b1)2

Slope Formula

The slope or inclination of a line can be determined in two ways. First is by knowing the angle extended by the line; if we know the angle, the slope is equal to tan θ. If the angle of a line is unknown, but the coordinates of any two points on the line are given then, slope equals to (b2-b1)/(a2-a1), where (a1, b1) are coordinates of the first point and (a2, b2) are coordinates of the second point

Section Formula in Coordinate Geometry

Let us assume that a line segment AB with points A (a1, b1) and B (a2, b2) is divided by another point C (a3, b3) in such a way that the ratio of two segments of the line is m:n. To find the value of this point, we use a section formula. It is given as

C (a3, b3) = {(m.a2 + n.a1)/(m+n), (m.b1 + n.b2)/(m+n)}

Mid-Point Formula

The mid-point formula is the special case of the section formula. Since the midpoint divides a line segment into equal parts, the ratio between the line segments is 1:1. There replace the value of m:n to 1:1 in the section formula, and we get the mid-point formula.

C (a3, b3) = {(a2 + a1)/2, (b1 + b2)/2}

Area of a Triangle Coordinate Geometry Formula

The coordinate geometry formula can be used to find the area of all triangles, irrespective of their types. For example, the area of a triangle with the vertices X (a1, b1), Y (a2, b2), and Z (a3, b3) is obtained from the following formula.

Area of a Triangle = ½ {a1(b2 – b3) + a2 (b3 – b1) + a3 (b1 – b2)}

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