Coordinate Geometry Formulas: Concept, Examples and FAQs

Coordinate Geometry Formulas

The coordinate geometry formulas include distance formula, slope formula, midpoint formula, the angle between lines formula, section formula, and the equation of a line formula. 

Before learning about these formulas, let us have a look at the concept of coordinate geometry.

What Is Coordinate Geometry?

Coordinate geometry is a representation of geometric forms on a two-dimensional plane. Before coming to formulas, we will understand the concept of a coordinate plane and the coordinates of a point.

Coordinate Plane

The cartesian plane consists of 2 number lines, x-Axis (Horizontal) and y-Axis (Vertical), perpendicular to each other. It is known as a coordinate plane. These x and y axes divide the cartesian plane into four quadrants, and the point of intersection of these two lines is the origin. 

Any random point on the coordinate plane is represented by a point (x, y), where the x value is the distance of the point from the x-axis, and the y value is the distance of the point from the y-axis.

The coordinates of origin O are (0, 0).

Any point represented in the first quadrant has both positive values, and its coordinates are in the form of (x, y).

Points represented in the second, third, and fourth quadrant have coordinates of the form (-x, y), (-x, -y), and (x, -y) respectively.

If two points A and B coordinates are \left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right), then the distance between these points is given as

d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}

Slope Formula

The slope of a line is the ratio of vertical change or inclination of the line. The slope of a line inclined at an angle θ with the positive x-axis is given by m = tan θ. 

The formula for the slope of a line joining the two points A\left(x_{1}, y_{1}\right) \text { and } B\left(x_{2}, y_{2}\right)is

m=\frac{\left(y_{2}-y_{1}\right)}{\left(x_{2}-x_{1}\right)}

Mid-Point Formula

For finding the coordinates of the mid-point of the line joining the points \left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)we use the formula

(x, y)=\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)

Section Formula: Coordinates of The Point Which Divides a Line into m:n Ratio

The section formula in coordinate geometry represents the coordinates of the point that divides a line segment joining the points A\left(x_{1}, y_{1}\right) \text { and } B\left(x_{2}, y_{2}\right)in the ratio m: n. We can find the coordinates of the point using the formula given below.

(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n},\frac{m y_{2}+n y_{1}}{m+n}\right)

Area of a Triangle Formula in a Coordinate Plain

The area of a triangle in a cartesian plane whose vertices are \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \text { and }\left(x_{3}, y_{3}\right) is given by the formula. 

\text { Area of the triangle }=\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|

The Angle Between Two Lines Formula:

Consider two straight lines A and B, having slopes m_1\text{ and }m_2, respectively.

If the angle between these lines is ‘θ’, then the angle between them can be calculated using the formula

\tan \theta=\frac{\left(m_{1}-m_{2}\right)}{\left(1+m_{1} m_{2}\right)}

Equation of a Line Formula in a Cartesian Plane

The equation of a line in a coordinate plane can be expressed in different ways. Some of them are given below.

(i) General Form

The general form of a line in a cartesian plane is represented by Ax + By + C = 0.

(ii) Slope intercept Form 

Let a line passes through a point whose coordinates are (x, y), ‘m’ be the slope of the line, and ‘c’ be the y-intercept, then the equation of a line formula is given by:

                                                             y = mx + c

Examples

1. Find the distance between two points P (4,5) and Q (-3,8)?

Ans: We know that the coordinate geometry distance formula is given by

d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}

\text { Here } x_{1}=4, y_{1}=5, x_{2}=-3, y_{2}=8

d =\sqrt{(-3-4)^{2}+(8-5)^{2}}
=\sqrt{(-7)^{2}+(3)^{2}}
=\sqrt{49+9}
=\sqrt{58}

Therefore, the distance between points is √58 units.

2. What is the equation of a line having a slope of (-4) and y-intercept of 3?

Ans:

Given,
Slope (m) = -4 and
y-intercept (c) = 3
Equation of slope intercept form of the line is given by the formula
y = mx + c
y = (-4)x + 3
y = -4x + 3
4x + y = 3
Therefore, the equation of the line is 4x + y = 3