Coordinate Geometry - Distance between Two Points

Coordinate Geometry - Distance between Two Points

1. The distance between two points which lie on the X-axis: The distance between the points A (x₁,0) and B (x₂, 0) is AB = |x₁-x₂|

2. The distance between two points which lie on the Y-axis : The distance between the points A (0, y₁) and B (0,y₂) is AB = |y₁-y₂|
        
3. The distance between two points which lie on either of the coordinate axis :    
    The distance between the points A (x, 0) and B (0, y) is
    From the diagram, it is clear that OA = x, OB = y
Clearly, \( \Delta ABC \) is a right angled triangle, by pythogorus theorem. We have 
\( AB^2 = OA^2 + OB^2 \)
\( = x^2 + y^2 \)
\(\therefore AB = \sqrt {x^2 + y^2 } \)
        
4.  Let (x₁, y₁), B(x₂, y₂) be any two points on a line not parallel to the axes. From the adjacent figure we have the right angle triangle ABC.

AB² = AC² + BC² But AC = x₂-x₁, BC = y₂-y₁
 \( \therefore AB^2 = \left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 \)  \( \therefore AB = \sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 } \)         
Note : The distance to the point A(x₁, y₁) from origin is \( \sqrt {x_1^2 + y_1^2 } \) 

Ex 1: Find the distance between the points (1, 2) and (3, 2) 
Sol:    Let A = (x₁, y₁) = (1, 2)
                B = (x₂, y₂) = (3, 2)

\(\therefore\) Distance between A and B = \( \sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 } = \sqrt {\left( {3 - 1} \right)^2 + \left( {2 - 2} \right)^2 } \)
                                                   \( = \sqrt {\left( 2 \right)^2 + \left( 0 \right)^2 } = 2\text{ }units \)

Coordinate Geometry-Distance between two points

Coordinate Geometry-Distance between two points

7.    Types of Quadrilaterals 
The quadrilateral formed by A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) and D(x₄, y₄) is
    Parallelogram when 
    (i)    Mid point of AC = mid point of BD
    A parallelogram ABCD is    
    Rhombus when
    i)     AB = BC             ii) AC \( \ne \) BD
    Rectangle when
    i)     AB  BC               ii) AC = BD
    Square when
    i)    AB = BC              ii) AC = BD