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 (x1, y1) and Q (x2, y2) 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(x1, y1) and B(x2, y2) 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(x1, y1) and B(x2, y2) 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 mid-point of the line joining A(x1, y1) and B(x2, y2) 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 videos|457 docs|77 tests
|
1. What is the Distance Formula in Coordinate Geometry? |
2. How is the Mid-Point 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
|