Coordinate Geometry: Cartesian Plane, Formulas and Examples

The Cartesian coordinate system is a mathematical branch that describes how to represent a point in the n-dimensional coordinate plane in a unique way. Rene Descartes, a French philosopher and mathematician, proposed the Cartesian system theory in the 17th century. The link between Euclidean geometry and algebra was established by this Cartesian coordinate system, which revolutionized the study of mathematics. The Cartesian coordinate system is used to depict lines, curves, and geometric shapes in the n-dimensional plane and is the cornerstone of analytical geometry. We'll discuss the Cartesian system in depth, including solved examples, formulas, coordinates, and graph plotting.

Key Takeaways: geometry, coordinates, Cartesian plane, quadrants, geometry graph, Cartesian distance, Cartesian midpoint

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Cartesian System

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The Cartesian system is the system that is used to label points in a plane and the number line gives rise to the Cartesian form. To comprehend the Cartesian coordinate system, we must have a thorough understanding of the number line. We have the following defined parameters in this system:

• The X-axis and Y-axis are two perpendicular lines.
• The plane is known as the Cartesian or coordinate plane, and the two lines X and Y, when combined, are known as the system's coordinate axes.
• The plane is divided into four quadrants by two coordinate axes.
• The Cartesian System's zero is the place where axes connect. The letter O will be used to represent this point. The origin's coordinates are written as (0, 0).
• To specify the position of any point P in the plane, we must first measure the distance x along X and then the distance y parallel to Y to get from O to P. Negative distances are possible.

The video below explains this:

Coordinate Geometry Detailed Video Explanation:

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Cartesian Coordinates

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The x-coordinate of a point in the Cartesian system is its perpendicular distance from the y-axis. It is measured along the x-axis, which is positive in one direction and negative in the opposite. The x-coordinate is called abscissa.

The y-coordinate of a point in the Cartesian system is its perpendicular distance from the x-axis. The y-axis is used to calculate it. The ordinate refers to the y-coordinate.

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Quadrants

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The quadrants divide the Cartesian plane into four equal pieces. These are numbered I, II, III, and IV in an anticlockwise sequence, beginning at the upper right and proceeding around in a clockwise orientation.

Points of Quadrants

I Quadrant: (+, +) It has a positive X coordinate and a positive Y coordinate.

II Quadrant: (-, +) It has a negative X coordinate and positive Y coordinate.

III Quadrant: (-, -) It has a negative X coordinate and negative Y coordinate.

IV Quadrant: (+, -) It has a positive X coordinate and a negative Y coordinate.

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Plotting the Graph

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Using the coordinate axes, it is simple to plot points on a given graph on the plane with an ordered number of pairs. If the coordinates (x,y) are given for a point P, then x is the abscissa and y is the ordinate.

The coordinate points will define the Cartesian plane's location. The horizontal axis is represented by the first point in the coordinates (x), and the vertical axis is represented by the second point in the coordinates (y).

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Formulas

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The formulas of the Cartesian coordinate system make it easy to prove the numerous properties of lines, curves, and planes in two and three dimensions. The distance formula, slope formula, midpoint formula, section formula, equations of a line in two and three dimensions, equations of curves, and equations of a plane are all part of the Cartesian coordinate system. The following are a few of these formulas:

• Coordinate Distance Formula- The following is the formula for calculating the distance between two points (x1,y1) and (x2,y2):
  D= √(x2-x1)²+ (y2-y1)²

• MidPoint Formula- The following is the formula of the midpoint between the two points (x1,y1) and (x2,y2):
  (x,y) = (x1+x2/2 , y1+y2/2 )
• Section Formula- The section formula is useful for determining the coordinates of a point on a line segment that splits the line segment connecting the points (x1,y1) and (x2,y2) in the ratio m:n :
  (x,y) = (mx2+nx1/m+n, my2+ny1/m+n)

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Things to Remember

• The X-axis and Y-axis are two perpendicular lines.
• The plane is known as the Cartesian or coordinate plane, and the two lines X and Y, when combined, are known as the system's coordinate axes.
• The plane is divided into four quadrants by the two coordinate axes.
• The Cartesian System's zero is the place where the axes connect. The letter O will be used to represent this point. The origin's coordinates are written as (0, 0).
• To specify the position of any point P in the plane, we must first measure the distance x along X and then the distance y parallel to Y to get from O to P. Negative distances are possible.
• I Quadrant: (+, +) It has a positive X coordinate and a positive Y coordinate.
• II Quadrant: (-, +) It has a negative X coordinate and a positive Y coordinate.
• III Quadrant: (-, -) It has a negative X coordinate and negative Y coordinate.
• IV Quadrant: (+, -) It has a positive X coordinate and a negative Y coordinate.
• Coordinate Distance Formula= D= √(x2-x1)²+ (y2-y1)²

• MidPoint Formula= (x,y) = (x1+x2/2 , y1+y2/2 )

• Section Formula= (x,y) = (mx2+nx1/m+n, my2+ny1/m+n)

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