Question

state and Prove the distance between two points in the plane​

Answer 1

Step-by-step explanation:

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, a2 +b2+=c2 a2 +b2 = c2, is based on a right angle where a and b are the length of the legs adjacent to the right angle, and C is the length of the hypotenuse.

Answer 2

√(  (x₂ - x₁)² +  (y₂ - y₁)² )  is the distance between  two points A & B in the plane where A  = (x₁ , y₁   )    & B = (x₂ , y₂)

Step-by-step explanation:

Let say Two points  are

A  = (x₁ , y₁   )    & B = (x₂ , y₂)

distance between two points in the plane​ A & B = √(  (x₂ - x₁)² +  (y₂ - y₁)² )

proof :

Lets draw a line parallel to x axis  passing though A

so slope of line would be zero

Hence  y - y₁  = 0(x - x₁ )

=> line would be  y =  y₁

now draw a  line parallel to y axis passing through point B

line would be  x  = x₂

intersection of both Lines at C would be ( x₂ , y₁)

now ACB will be a right angle triangle at C  ( as x axis & y axis are perpendicular to each other)

now AC =  (x₂ - x₁)

& BC = (y₂ - y₁)

applying Pythagoras theorem

AB² = AC² + BC²

=> AB² =   (x₂ - x₁)² +  (y₂ - y₁)²

=> AB = √(  (x₂ - x₁)² +  (y₂ - y₁)² )

QED

Hence Proved

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