Coordinate Geometry is considered to be one of the most interesting parts of the geometrical mathematics. Here, in this topic, the concepts of coordinate geometry or Cartesian geometry will be explained with Coordinate Geometry Formula with examples. Let us begin learning!

**Coordinate Geometry Formula**

**Introduction to Coordinate Geometry**

Coordinate Geometry also is known as analytic geometry that describes the link between geometry and algebra using graphs and involving curves and lines. It provides geometrical aspects in Algebra and hence enables to solve the complex geometrical problems. It is a part of geometry where the position of points on the plane is represented using an ordered pair of numbers.

Coordinate geometry or Cartesian geometry is defined as the study of geometry using the coordinate points. With the help of coordinate geometry, various computations in geometry can be done easily. Some of these are:

- Finding the distance between two given points.
- Dividing the line in given ratio m:n.
- Finding the mid-point of a line.
- Calculating the area of a triangle in a Cartesian plane, etc.

**Coordinate Geometry Formula**

**(1) Distance Formula: To Calculate Distance Between Two Points:**

Let the two points be A and B, having coordinates to be (x_1,y_1) and (x_2,y_2) respectively.

Thus, the distance between two points is-

distance = \(\sqrt {( {x_1 – x_2 } )^2 + ( {y_1 – y_2 } )^2 }\)

**(2) Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points:**

Consider the same points A and B, having coordinates to be \((x_1,y_1)\) and \((x_2,y_2)\) respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of this point are –

\(\frac {x_1+x_2}{2}\) , \(\frac {y_1+y_2}{2}\) ,

**(3) Angle Formula: To Find The Angle Between Two Lines:**

Consider two straight lines and , with given slopes as m_1 and m_2 respectively.

Let “θ” be the angle between these two lines, we can then represent the angle between them as-

tanθ= \(\frac{ m1–m2 }{1+m1 \times m2}\)

**(4) Section Formula: To Find a Point which divides a line into m:n Ratio using: **

Consider a two straight lines having coordinates \((x_1,y_1)\) & \((x_2,y_2)\) respectively. Let a point which divides the line in some ratio as m:n, then the coordinates of this point are-

- When the ratio m:n is internal:

\(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\)

- When the ratio m:n is external:

\(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\)

**(5) Area of a Triangle in Cartesian Plane:**

We can compute the area of a triangle in Cartesian Geometry if we know all the coordinates of all three vertices. If coordinats are \((x_1,y_1)\),\((x_2,y_2)\) and \((x_3,y_3)\) then area will be:

Area =\(\frac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]\)

## Solved Examples

Q.1: Find the distance between two points with coordinates (4,5) and (-3,8).

Solution: Here, points are (4,5) and (-3,8)

Thus \(x_1\)=4

\(Y_1\)=5

\(X_2\)=-3

\(Y_2\)=8

Now distance formula is,

distance = \(\sqrt {( {x_1 – x_2 } )^2 + ( {y_1 – y_2 } )^2 }\)

Putting all known values, we get

d = \(\sqrt{49+9}\)

d= \(\sqrt{58}\)

d= 7.61

Thus distance between he points is 7.61.

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