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Coordinate Geometry

Cartesian Coordinate System

In the Cartesian coordinate system, there is a Cartesian plane which is made up of two number lines which are perpendicular to each other, i.e. x-axis (horizontal) and y-axis (vertical) which represents the two variables. These two perpendicular lines are called the coordinate axis.

  • The intersection point of these two lines is known as the center or the origin of the coordinate plane. Its coordinates are (0, 0).

  • Any point on this coordinate plane is represented by the ordered pair of numbers. Let (a, b) is an ordered pair then a is the x-coordinate and b is the y-coordinate.

  • The distance of any point from the y-axis is called its x-coordinate or abscissa and the distance of any point from the x-axis is called its y-coordinate or ordinate.

  • The Cartesian plane is divided into four quadrants I, II, III and IV.

Equation of a Straight Line

An equation of line is used to plot the graph of the line on the cartesian plane.

The equation of a line is written in slope intercept form as

y = mx +b

where m is the slope of the line and b is the y intercept.

To find the slope of the line first we need to convert the equation in slope intercept form then we can get the slope and y intercept easily.

Distance formula

The distance between any two points A(x1,y1) and B(x2,y2) is calculated by

Example

Find the distance between the points D and E, in the given figure.

Solution

This shows that this is the same as Pythagoras theorem. As in Pythagoras theorem

Distance from Origin

If we have to find the distance of any point from the origin then, one point is P(x,y) and the other point is the origin itself, which is O(0,0). So according to the above distance formula, it will be

Section formula

If P(x, y) is any point on the line segment AB, which divides AB in the ratio of m: n, then the coordinates of the point P(x, y) will be

Mid-point formula

If P(x, y) is the mid-point of the line segment AB, which divides AB in the ratio of 1:1, then the coordinates of the point P(x, y) will be

Area of a Triangle

Here ABC is a triangle with vertices A(x1, y1), B(x2, y2) and C(x3, y3). To find the area of the triangle we need to draw AP, BQ and CR perpendiculars from A, B and C, respectively, to the x-axis. Now we can see that ABQP, APRC and BQRC are all trapeziums.

Area of triangle ABC = Area of trapezium ABQP + Area of trapezium APRC – Area of trapezium BQRC.

Therefore,

Remark: If the area of the triangle is zero then the given three points must be collinear.