1, formula: x+y = a+b

This formula is actually a mathematical model, and the coordinates of a point can be expressed by two variables X and Y in rectangular coordinate system. Where x represents the position of the point on the X axis and y represents the position of the point on the Y axis. This formula tells us that given the values of x and y, we can find the position of the corresponding point in the rectangular coordinate system.

2. Formula: distance from point to origin = √(x? + y？ )

This formula is used to calculate the distance from a point to the origin of a rectangular coordinate system. Where x and y are the coordinates of the point. This formula uses the principle of Pythagorean theorem, that is, the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right-angled sides. Using this formula, we can find the distance from any point to the origin.

3. Formula: the distance from (x 1, y 1) to (x2, y2) = √[(x2-x 1)? + (y2 - y 1)？ ]

This formula is used to calculate the distance between two points. Where (x 1, y 1) and (x2, y2) are the coordinates of two points. This formula also uses the principle of Pythagorean theorem. The distance between two points can be obtained by calculating the coordinate difference between two points on the X axis and the Y axis, then adding squares, and finally finding the square root.

Similarities and differences between rectangular coordinate system and polar coordinate system;

Difference:

1. Definition: Cartesian coordinate system describes the position of a point by defining the number axes perpendicular to each other on the plane and having a common origin, while polar coordinate system describes the position of a point by the distance from the origin to the point (polar diameter) and the angle from the positive direction of the X axis to the counterclockwise rotation of the line segment (polar angle).

2. Coordinate representation: In rectangular coordinate system, the position of a point is determined by its coordinates (x, y) on the X axis and Y axis, while in polar coordinate system, the position of a point is determined by its polar diameter r and polar angle θ.

3. Coordinate transformation: the transformation relationship between the two coordinate systems is clear. The conversion formula from rectangular coordinates to polar coordinates is r=√(x? +y？ )，θ= arctan(y/x)； The conversion formula from polar coordinates to rectangular coordinates is x=rcos(θ) and y=rsin(θ).

Similarities:

1. Description of position: Cartesian coordinate system and polar coordinate system are both coordinate systems used to describe the position of a point on a plane.

2. Origin: Both coordinate systems use the origin as a reference point to determine the positions of other points.