Step-by-Step Learning
Learn how to use formulas to solve problems in co-ordinate geometry.
Example 1: Distance Formula
Find the distance between two points A(x₁, y₁) and B(x₂, y₂).
Concept: The distance formula is derived from the Pythagoras Theorem and is used to find the distance between any two points in a coordinate plane.
Formula: The distance between points A(x₁, y₁) and B(x₂, y₂) is given by:
Distance AB = square root of ((x₂ - x₁)² + (y₂ - y₁)²).
Example: Find the distance between A(2, 3) and B(5, 7).
x₁ = 2, y₁ = 3, x₂ = 5, y₂ = 7.
Distance AB = square root of ((5 - 2)² + (7 - 3)²)
Distance AB = square root of ((3)² + (4)²)
Distance AB = square root of (9 + 16)
Distance AB = square root of 25
Distance AB = 5 units.
Example 2: Section Formula (Internal Division)
Find the coordinates of the point P(x, y) that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂.
Concept: The section formula is used to find the coordinates of a point that divides a line segment in a given ratio.
Formula: The coordinates of point P(x, y) are given by:
x = (m₁x₂ + m₂x₁) / (m₁ + m₂)
y = (m₁y₂ + m₂y₁) / (m₁ + m₂)
Example: Find the coordinates of the point that divides the segment joining A(1, 2) and B(4, 5) in the ratio 2:1.
x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 5, m₁ = 2, m₂ = 1.
x = (2*4 + 1*1) / (2 + 1) = (8 + 1) / 3 = 9 / 3 = 3
y = (2*5 + 1*2) / (2 + 1) = (10 + 2) / 3 = 12 / 3 = 4
The point is (3, 4).
Example 3: Area of a Triangle
Find the area of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃).
Concept: The area of a triangle in the coordinate plane can be calculated using the coordinates of its vertices.
Formula: The area of Triangle ABC is given by:
Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
(The vertical bars indicate the absolute value, as area is always non-negative).
Example: Find the area of the triangle with vertices A(1, 1), B(3, 4), and C(5, 2).
x₁ = 1, y₁ = 1, x₂ = 3, y₂ = 4, x₃ = 5, y₃ = 2.
Area = 1/2 * |1(4 - 2) + 3(2 - 1) + 5(1 - 4)|
Area = 1/2 * |1(2) + 3(1) + 5(-3)|
Area = 1/2 * |2 + 3 - 15|
Area = 1/2 * |-10|
Area = 1/2 * 10
Area = 5 square units.
Note: If the area calculated is 0, the three points are collinear (lie on the same straight line).
Practice Mode
Enter problem details to calculate distance, section point, or area of a triangle.
Examples: "distance (2,3) and (5,7)", "section (1,2) and (4,5) ratio 2:1", "area (1,1), (3,4), (5,2)".