Table of contents  
What is a Coordinate System?  
Distance Formula  
Section Formula  
Mid Point Formula  
Some Solved Questions 
A Coordinate System is a mathematical framework used to determine the position or location of points in space. It provides a way to describe the position of objects or points using numerical values called coordinates.
The distance between two points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) is given by:
This is also known as the Distance Formula.
Note: The distance of any point P(x,y) from the origin O(0,0) is given by:
Example 1: Find the distance between the points D and E, in the given figure.
Solution:
Example 2: What is the distance between two points (2, 3) and (4, 5) using the distance formula?
Sol: The distance formula is used to calculate the distance between two points in a coordinate plane. It is given as:
d = √[(x2  x1)² + (y2  y1)²]
Using this formula, we can find the distance between the points (2, 3) and (4, 5) as follows:
d = √[(4  2)² + (5  3)²]
d = √[(6)² + (2)²]
d = √[36 + 4]
d = √40
d = 6.32 (approx.)
Therefore, the distance between the points (2, 3) and (4, 5) is approximately 6.32 units.
Let P (x,y) be a point on the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) such that it divides AB internally in the ratio m:n. The coordinates of the point P are given by
This is known as the Section Formula.
Note:
(i) If the point P divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) internally in the ratio k:1, its coordinates are given by:
Example 2: In what ratio does the point (2, 5) divide the line segment joining the points A(3, 5) and B(4, 9).
Sol: Let the ratio be λ : 1
We have put m = λ and n = 1
or
But, coordinates of point is given as p(2,5)But, coordinates of point is given as p(2,5)
4λ  3 = 2(λ + 1)
⇒ 4λ = 2λ + 2 + 3⇒ 2λ = 5
⇒ λ = 5/2The required ratio is 5:2.
The midpoint of the line joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given as
Example 3: Suppose we have two points A(2, 4) and B(6, 8). We want to find the midpoint of the line segment AB.
Using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
= ((2 + 6) / 2, (4 + 8) / 2)
= (8 / 2, 12 / 2)
= (4, 6)
Therefore, the midpoint of the line segment AB is M(4, 6).
Q1: Find the distance between the points (3, 5) and (2, 1) using the distance formula.
Solution:
Using the distance formula:
d = √[(x2  x1)² + (y2  y1)²]
Substituting the coordinates:
d = √[(2  3)² + (1  5)²]
d = √[(5)² + (6)²]
d = √[25 + 36]
d = √61
Therefore, the distance between the points (3, 5) and (2, 1) is √61 units.
Q2: Find the coordinates of the midpoint of the line segment joining the points (3, 2) and (5, 4).
Solution:
Using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Substituting the coordinates:
Midpoint = ((3 + 5) / 2, (2 + (4)) / 2)
Midpoint = (2 / 2, 2 / 2)
Midpoint = (1, 1)
Therefore, the midpoint of the line segment joining (3, 2) and (5, 4) is (1, 1).
124 videos457 docs77 tests

1. What is the Distance Formula in Coordinate Geometry? 
2. How is the MidPoint Formula used in Coordinate Geometry? 
3. What is the Section Formula in Coordinate Geometry? 
4. How is the Centroid of a Triangle calculated using Coordinate Geometry? 
5. What is the Condition for Collinearity of Three Points in Coordinate Geometry? 

Explore Courses for Class 10 exam
