 In maths, coordinate geometry is one of the most important and interesting concepts at the secondary level. Coordinate geometry is the part of the geometry that uses two or more to specify the point. The position of the point or figure can be determined in a line or three-dimensional space. The concept of coordinate geometry is explained with definitions, formulas, examples here. In addition to this, we also provide worksheets on coordinate geometry to test your knowledge of this chapter.

## Topics in Co-ordinate Geometry Grade 9 | Coordinate Geometry List of Contents

### Introduction to Coordinate Geometry

Coordinate geometry is determined as the study of geometry using coordinate points. With the help of the points, we can find the distance between the two points, divide the lines into m:n ratio, find the midpoint, calculate the area of a triangle in the cartesian plane, and so on.

 Definition of Coordinate geometry It is a part of the geometry where the position of the point is determined using coordinates. What is meant by coordinate? Coordinates are nothing but the set of values that helps to show the exact point in the coordinate plane. Distance Formula The distance formula is used to find the distance between two points A(x1, y1) and B(x2, y2). What is meant by Coordinate Plane? A coordinate plane is a two-dimensional plane that is formed by the intersection of two perpendicular lines that is x-axis and y-axis. Section Formula Section formula is used to divide any line into two parts m:n ratio Mid-point theorem Mid-point theorem is used to find the coordinates at which a line is divided into two parts.

### What is a Co-ordinate and a Co-ordinate Plane?

Coordinate and coordinate plane sounds the same but they are different. A coordinate is the set of values that is used to show the exact point in the coordinate plane. Whereas a coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines that is x-axis and y-axis. The point where the two axes intersect is known as the origin. The location of the point (x, y) is known as the coordinates.
There are four quadrants in the graph they are, ### Equation of a Line in Cartesian Plane

Equation of the line can be represented as follows,
i. General Form:
The general form of a line is given as Ax + By + C = 0.
ii. Slope intercept form:
Let x, y be the coordinate of a point that passes through the line, m be the slope of the line, and c be the y-intercept.
y=mx + c
iii. Intercept Form of a line:
Consider a and b be the x-intercept and y-intercept of a line, then the equation of a line is represented as
y=mx + c

### Slope of a Line

The general form of the line is Ax + By + C = 0.
The slope can be found by converting this form to the slope-intercept form.
Ax + By + C = 0
By = − Ax – C
By = − Ax – C
y = -A/B x – C/B
Comparing the above equation with y = mx + c,
m = -A/B
By using this formula we can find the slope of a line from the general equation of a line.

### Coordinate Geometry Formulas and Theorems

Distance Formula:
The distance formula is used to find the distance between two points A(x1, y1) and B(x2, y2).
d = √(x2 – x1)² + (y2 – y1)²
where (x1, y1) and (x2, y2) are the coordinates of the plane.
Mid-point Theorem:
The mid-point theorem is used to find the coordinates at which a line is divided into two parts. Consider two points A and B having the coordinates (x1, y1) and (x2, y2)
M(x, y) = [(x1+x2)/2, (y1+y2)/2]
Angle Formula:
Consider two lines A and B, having their slopes to be m1 and m2 respectively.
Let “θ” be the angle between these two lines, then the angle between them.
tanθ = (m1-m2)/1+m1m2
Case 1: When the two lines are parallel to each other,
m1 = m2 = m
tanθ = (m1-m2)/1+m1m2 = 0
Thus θ = 0
Case 2: When the two lines are perpendicular to each other,
m1.m2 = -1
tanθ = (m1-m2)/1+m1m2
tanθ = (m1-m2)/1+(-1) = (m1-m2)/0 = undefined
θ = 90°

### Coordinate Geometry Example Questions

Example 1.
Find the distance of the point P (7, 8) from the x-axis.
Solution:
We know that,
(x, y) = (7, 8) is a point on the Cartesian plane in the first quadrant.
x = Perpendicular distance from the y-axis
y = Perpendicular distance from x-axis
Therefore, the perpendicular distance from x-axis = y coordinate = 8

Example 2.
Find a relation between x and y such that the point P(x, y) is equidistant from the points A (2, 4) and B (3, 6).
Solution:
Let P (x, y) be equidistant from the points A (2, 4) and B (3, 7).
Therefore AP = BP …[Given]
AP² = BP² …[Squaring both sides]
(x – 2)² + (y – 4)² = (x – 3)² + (y – 6)²
x² – 4x + 4 + y² – 8y + 16 = x² – 6x + 9 + y² -12y + 36
-4x -8y + 20 = -6x – 12y + 45
-4x -8y + 20 + 6x + 12y – 45 = 0
2x + 4y – 25 = 0
2x + 4y = 25
2x – 25 = -4y
2x – 25 = -4y is the required relation.

Example 3.
Three vertices of a parallelogram taken in order are (1,0) (2,1) and (2,3) respectively. Find the coordinates of the fourth vertex
Solution:
Consider D(x,y) as the fourth vertex.
Let A(1,0) B(2,1) C(2,3) and D(x,y) be the vertices of a parallelogram.
ABCD took each other.
Since the diagonal of a parallelogram bisects each other.
Coordinates of the mid point of AC = coordinates of the midpoint of BD.
(1 + 2/2 , 0 + 3/2) = (2 + x/2 , 1 + y/2)
(3/2 , 3/2) = (2 + x/2 , 1 + y/2)
2 + x/2 + 3/2
= 2 + x = 3
x = 3 – 2
x = 1
And
1 + y/2 = 3/2
1 + y = 3
y = 3 – 1
y = 2
Hence coordinates of the fourth vertex D(1,2)

Example 4.
Find that value(s) of x for which the distance between the points P(x, 3) and Q(7, 10) is 10 units.
Solution:
Given that
PQ = 10
PQ² = 10²
PQ² = 100
(7 – x)² + (10 – 3)² = 100
Using distance formula
(7 – x)² + (10 – 3)² = 100
(7 – x)² = 100 – 49
(7 – x)² = 51
(7 – x) = √51
7 – x = + or – 7.1
7 – x = 7.1 or 7 – x = -7.1
x = 7.1 – 7 or x = -7.1 – 7
x = 0.1 or x = -14.1

Example 5.
Find the centroid of the triangle whose vertices are (1,2) (3,4) (5,6).
Solution:
The centroid of the triangle whose vertices are (1,2) (3,4) and (5,6)
The centroid of the triangle ABC = (x1 + x2 + x3/2, y1 + y2 + y3/2)
The centroid of the triangle whose vertices A(1,2) B(3,4), C(5,6)
= (1+3+5/2 ,2+ 4 + 6/2)
(9/2, 12/2)
(9/2, 6).

### FAQs on Coordinate Geometry

1. How can you prepare and complete coordinate geometry?

Pay attention to the classes and practice the problems thoroughly. Make use of the links given on our page and clarify your doubts.

2. How Can You Define the Position of an Object on a Floor?

The students of 9th grade can define the position of an object on the floor by considering two adjacent walls as two coordinate axes.

3. What are the signs of the coordinates in the four quadrants of a cartesian plane?

The signs of the coordinates in the four quadrants of a cartesian plane are (+, +) in the first quadrant, (-, +) in the second quadrant, (-, -) in the third quadrant of a cartesian plane, and (+, +) in the fourth quadrant of a cartesian plane.