Coordinate Geometry P6

  About Coordinate Geometry

Let's discuss about what exactly Coordinate Geometry is?

Coordinate Geometry:   It is the branch of Geometry where we study about Geometry(like  distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc.) by using Coordinate System.

Note: In classical mathematics, coordinate geometry, also known as analytic geometry or Cartesian geometry.

Example1: look here...

Here P(5,4) is coordinate of  point P which is in 1st (I) Quadrant.

Abscissa:   It refers to the Horizontal  X -axis.  Here Abscissa is 5, since this is horizontal  X-axis and has 5 Unit distance .

Ordinate: It refers to the Vertical Y -axis.  Here Ordinate is 4, since this is vertical Y-axis and has 4 Unit distance .

Example2:

In Coordinate System we can represent any finite point in this Plane. here above mentioned L,M and N points.

Now...

If you are confuse about Quadrant Sign then look General form here.....

Quadrant I  has (+,+) means (x, y) respectively values. here  (+,+) = = (x, y).  So we can define all other Quadrant same way.

How to find Distance between Two Points

1st let's remind Pythagorean theorem,

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“

a= side1 

b= side2

c= hypotenuse side

We use Same concept to find distance between two points, let's see...

here let's say,   

(x1, y1)  and (x2, y2) are two coordinate points as given above, so the Formula is like..

Let's see Some Examples of same type to find Distance.

                                                    Another Example

 

                                                    Another Example

Learn Section Formula

If a point $P = (x , y)$ divides the line joining two points $A = (a , b)$ and $B = (c , d)$ in the ratio $m : n$ internally, then the coordinates of $P$ are given by

$$P~(x , y) = \left (\dfrac{c \cdot m + a \cdot n}{m + n} , \dfrac{d \cdot m + b \cdot n}{m + n} \right).

If $P$ divides $AB$ externally in the ration $m : n$, then

$$P~(x , y) = \left(\dfrac{c \cdot m - a \cdot n}{m - n} , \dfrac{d \cdot m - b \cdot n}{m - n} \right).

Note: Sometimes we use m/n=k(constant), so desired ratio is k:1.

how?

original ratio given as  m:n

m:n=m/n : n/n=k:1    (Divide both side by n)

then apply section formula it's easy to solve problem, but same answer.

Finding Area of Tringle in Coordinate Geometry

We can find area of any tringle by using Coordinate Points.

let's see here...

Look above Figure each vertices has its own Coordinate points so we can easily find area of Tringle.

Formula to Find Area of Tringle

Area Of  Tringle

Note: Area of Triangle can't be Negative.

Note: If Area of Triangle is Zero means all points are at same line, called Collinear Points.