Two-Dimensional Coordinate Geometry: Coordinate Geometry in Two-Dimensional Plane

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Two-dimensional coordinate geometry deals about the coordinates which are represented in a coordinate plane. A coordinate plane has two axes, the one which is horizontal is known as and the one which is vertical is known as A point is represented in the as shown below.

Coordinate Geometry in Two-Dimensional Plane

Coordinate Geometry in Two-Dimensional Plane

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is known as origin whose coordinate is

The perpendicular distance of from and is and respectively.

Coordinate Geometry in Two-Dimensional Plane

For example, the point is away from the measured along the positive and away from measured along the positive

The points having coordinate as lie on the and points having coordinate as lie on the

For example, the points lie on and the points lie on

Distance between two points:

Consider two points and in an plane.

Then the distance between and is,

For example, distance between the points and is,

Similarly, distance between a point from the origin is,

Reflection of a point across the X-axis

Reflection of a point across the is which is found by changing the sign of the coordinate of

Coordinate Geometry in Two-Dimensional Plane

Coordinate Geometry in Two-Dimensional Plane

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Reflection of a point across the Y-axis

Reflection of a point across the is which is found by changing the sign of the coordinate of .

Coordinate Geometry in Two-Dimensional Plane

Coordinate Geometry in Two-Dimensional Plane

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Section Formula:

Consider two points and in an plane.

is a point which divides the line segment internally in the ratio

Then, the coordinate of is,

the coordinate of is,

If then is the mid-point of the line segment Then coordinates of the point is,

If the point is dividing the line segment externally in the ratio

Then, the coordinate of is,

they-coordinate of is,

Example: Find the coordinates of the point which divides the line segment joining and

in the ratio

Let be the point which divides in the ratio then

Therefore origin divides the point in the ratio

Area of a triangle:

The area of the triangle whose vertices are is

If the area of a triangle whose vertices are \ is zero, then the three points are collinear.

Example 1: Find the distance between the point .

Solution:

Here given two points,

We have to find out the distance between and

We have formula for the distance

We put the values,

First, we take subtraction in the root,

Take the square,

Take the sum in the root,

Take the square root of 100 and we get,

Example: 2 Find the coordinates of the point which divides the line segment joining and in the ratio

Solution:

Here given, two points, and and ratio .

Let be the point which divides in the ratio then

For find of , formula,

For find of , formula,

Therefore, divides the point in the ratio