Coordinate geometry – Formulas, Solved Examples

Coordinate Geometry 

Coordinate geometry is used for the representation of two-dimensional figures in the Cartesian plane. In a Cartesian plane ‘X’ and ‘Y,’ refer to the perpendicular distance of a point from the Y and X-axis respectively. A geometrical figure is represented with the help of points on a coordinate plane in the form of (x,y), where x refers to the x coordinate and y refers to the y coordinate of any point.

Formula:

The different formulas associated with coordinate geometry are given below:

1. Section formula: 

Let ‘P’ (x,y) be an internal point dividing point ‘A’ with coordinates \left(x_{1}, y_{1}\right)and point ‘B’ with coordinates \left(x_{2}, y_{2}\right)in the ratio m:n, then the coordinates of P is given by,

2. Distance between 2 points:

The distance between two points ‘A’ with coordinates \left(x_{1}, y_{1}\right)and point ‘B’ with coordinates \left(x_{2}, y_{2}\right)is given by,

3. The centroid of a triangle:

Let ΔABC be a triangle with vertices A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \text {, and } C\left(x_{3}, y_{3}\right)then the centroid of the triangle is given by,

Solved Examples:

Q1) What is the centroid of a triangle with vertices A (1, 3), B(2, 2), and C(3, 1)?

Ans: The centroid of a triangle with vertices A (1, 3), B(2, 2), and C(3, 1) is:

X = (1+2+3)/3 = 2

Y = (3+2+1)/3 = 2

So the coordinate of centroid is (2,2).

Q2) What is the distance between points A (8, 8) and B (5, 4)?

Ans: The distance between the points A (8, 8) and B (5, 4) is:

D=\sqrt{\left[(8-5)^{2}-(8-4)\right]^{2}}=\sqrt{25}=5 \text { units }

Q3) What is the coordinate of point C dividing the points A(1,1) and B(3,3) in the ratio 1:1 on a coordinate plane?

Ans: The coordinate of point C dividing the points A(1,1) and B(3,3) in the ratio 1:1 on a coordinate plane is: 

x=\frac{(1 \times 1)+(1 \times 3)}{2}=2 \text { and } y=\frac{(1 \times 1)+(1 \times 3)}{2}=2

So the coordinates of C are (2,2) on the coordinate plane.

Uses of Coordinate Geometry

1. Coordinate geometry used in navigation devices.
2. It is used in various editing software for drawing different kinds of shapes and edits.
3. Pixels of a photo position themselves using coordinate geometry in the camera.
4. It is used in drawing bar graphs, histograms, etc.