Area of Triangle and Collinearity

I dedicate this post to area of triangle drawn over Cartesian plane means triangle having vertices as coordinates, to find their area.

Area of Triangle in coordinate geometry

I guess you have studied " To find the area Of Triangle" with formula 

Ar (∆ ABC) = ½ × Base × Height

no doubt, this is one of the best and easy method to find area of any triangle If we have proper values of dimensions.

But if the triangle drawn over Cartesian plane, then triangle having all the three vertices as
coordinates A (X,Y), B (x, y) & C (x', y')

then area of this triangle can be formulated as 

Ar (∆ABC) =  ½ [ X (y - y') + x ( y' - Y) + x' ( Y - y) ]

for an example:-

if,  A (2,4), B (5,-6), C( -3,-5)

Then
Ar (∆ ABC) = ½ [ 2{-6 - (-5)} + 5{-5 -4} + (-3){4 -(-6)} ]

                 = ½[ 2(-6+5) + 5(-9) -3(4+6)]
                 = ½ [ 2( -1) -45 -30 ]
                 = ½ [ -2 -75 ]
                 = ½ [- 77]
                 = [-77/2]
     absolute value will be taken as area can not be in negative i.e.

 Ar (∆ABC) = 77/2   sq.unit

same area can be evaluated from the basic formula i.e.
area of Triangle =½ × base × height
base and height can be calculated from distance formula.

for Distance formula check out Distance formula- How to apply

Collinearity ( Points on same line):-

I have explained collinearity earlier in my other post ( you can read it here Collinearity)

but there is one more way to prove, for any three coordinates, are collinear or not.

For this we can use the same formula which we have used for finding area of any triangle, but the difference is this time the formula will placed equal to zero i.e.

Three vertices as  coordinates A (X,Y), B (x, y) & C (x', y')

then for collinearity the value of above formula i.e.

         ½ [ X (y - y') + x ( y' - Y) + x' ( Y - y) ] should be equal to zero i.e.

 ½ [ X (y - y') + x ( y' - Y) + x' ( Y - y) ]  = 0

maybe some curious minds can rise the question " why this happens - formula become equal to zero"

for those I have my own perspective which I think to share i.e.

when we are looking to find area of Triangle, one simply puts the respective values into formula and can surely gets the right answer,

but when we talk about Collinearity ( points on same line) which means those particular points aren't occupying​ area that's why we'll have to put above mentioned formula equal to zero or simply if we'll have to prove for some points whether they are collinear or not, we'll have to put the values into formula and if calculation comes to zero that means points are collinear.

Okay..!!

so this is all for today,

I hope you've got a good knowledge. Now over to you, Share if you want to add something that I'm missing let us know by commenting I would like to hear from you.
keep learning.