Distance between Two Points in a Plane
Distance between Two Points in a Plane:
Cartesian Coordinate System:
 Concept: The distance between two points in a plane can be calculated using the coordinates of those points.
 Formula: The distance formula between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is given by: $d=\sqrt{({x}_{2}{x}_{1}{)}^{2}+({y}_{2}{y}_{1}{)}^{2}}$
Steps to Calculate Distance:
 Identify Coordinates: Determine the coordinates of the two points given.
 Apply Formula: Plug the coordinates into the distance formula.
 Calculate: Use arithmetic operations (addition, subtraction, and squaring) as per the formula.
 Square Root: Calculate the square root of the sum of squares to find the distance.
Properties:
 NonNegative Value: The distance between two points is always nonnegative.
 Symmetry: The distance between point A and B is the same as between point B and A.
 Triangle Inequality Theorem: For any three points, the sum of the distances between any two points is greater than the distance between the remaining two points.
Applications:
 Geometry: Measuring lengths of line segments, diagonals of shapes.
 Physics: Calculating distances between objects or points in space.
 Engineering: Designing layouts, structures, and calculating dimensions.
Practical Examples:
Example 1: Distance between Two Cities:
 Coordinates: Represent cities as points on a map using latitude and longitude coordinates.
 Distance Calculation: Use formulas such as Haversine or Vincenty to find the distance between cities on the Earth's surface.
Example 2: Distance in 3D Space:
 Coordinates: Use threedimensional coordinates (x, y, z) to calculate distances between points in three dimensions.
 Distance Formula: Extends to 3D space using $d=\sqrt{({x}_{2}{x}_{1}{)}^{2}+({y}_{2}{y}_{1}{)}^{2}+({z}_{2}{z}_{1}{)}^{2}}$
Importance:
 Navigation: Finding distances between locations on maps and GPS systems.
 Mathematical Analysis: Critical in geometry, trigonometry, and calculus.
 RealWorld Applications: Used extensively in science, engineering, and various fields for measurement and calculations.
Example:
Consider two points, A and B, with coordinates A(3, 5) and B(2, 1). Find the distance between these two points.
Steps to Solve:

Identify Coordinates:
 Point A: ${x}_{1}=3,{y}_{1}=5$
 Point B: ${x}_{2}=2,{y}_{2}=1$

Apply Distance Formula: $d=\sqrt{({x}_{2}{x}_{1}{)}^{2}+({y}_{2}{y}_{1}{)}^{2}}$

Substitute Values: $d=\sqrt{(23{)}^{2}+(15{)}^{2}}$

Perform Arithmetic: $d=\sqrt{(5{)}^{2}+(6{)}^{2}}$ $d=\sqrt{25+36}$ $d=\sqrt{61}$

Final Answer: The distance between points A and B is $\sqrt{61}$ units.
Interpretation:
The distance between points A and B in the given coordinate system is approximately $7.81$ units, calculated using the distance formula. This distance represents the length of the straight line connecting these two points on the plane.