Proving the Points to be Collinear? - Coordinate Geometry Video Lecture - Class 9

based on the area of the triangle we
have another question here prove that
the given points a I'm sorry I write
here twice a2 minus 2b minus 3/8 and C
minus 1 for our collinear points so
collinear points concept I think you're
very well familiar with this you know
this Co means together and linear means
line so the points which are together in
a line are collinear points so here we
have got ABC alright this is point a
this is point B and this is Point C all
right and if these all points are
together in a line suppose this is our
line all right
if these all points are together in a
line say point a point B and Point C if
these all points are together in line
then I can say that these are collinear
points and then I can prove it but how
can I so that these are together in a
line so if any three points are not
together in a line then what happens all
right this is in the together in the
line suppose like like here I'm gonna
say 1 2 3 I have three points right if
they are together in a line then they
will form a straight line if they are
not then what will happen if any one of
them is a little bit away from the line
like like this is the line and a little
bit oh it is so let me join this and I
can let me try this so what I'm getting
I'm getting a triangle right it is one
point here is one point here is one
point if they all all are in the line in
a straight line then they will form a
straight line right but if they are not
if any one of them is a little bit
shifted like here then what they will -
they will form a triangle all right and
when the triangles area when triangles
area will be 0 triangles area will be 0
when triangle will be like like like
initially there initially the triangle
was like this much bigger right and the
area was area was something X and now it
the triangle is getting smaller all
right now the area will be decreasing
and now this is two points and here is
now so triangle is even smaller and here
is the two
now the third point is in the same line
now these going to form what so this is
a now the three points all are in one
line and that time area is going to be
zero so what I mean to say if I'm gonna
write area of triangle ABC if I'm going
to be successful - so that area of the
triangle ABC equals zero then it means
I'm proving that these three points are
collinear if I will be successful - so
that area of triangle ABC is zero then
yes showing that these points are
collinear because only cone here it only
only three culinary point scan form the
triangle whose area is zero whose area
eg remains nothing
if I'm saying like I have 1 million
rupees in a bank account then what is it
man that I have in my account right if
I'm saying I have zero billion dollar in
my bank account then what does it mean I
have no money at all in my bank account
all right so zero billion dollar what
does that mean if I'm saying trillion
dollar also that does not mean if I'm
right if I'm saying zero trillion dollar
right trillion dollar so what finally I
don't have anything in my in my bank
account all right the same thing so area
of the triangle is zero so here there is
no possibilities to make the area to
make the triangle all right so area of
the triangle will be 0 1 only when there
is no triangle at all so let's find the
area of the triangle now all right so to
find the area of the triangle 1 by 2 now
ABC all right so first point - - - - - -
second point that is minus 3/8 and the
third point is minus 1/4 and you know
that we need to repeat this one more
time so it's 2 minus 2 we need to so
both side equal right but left hand side
as well as right hand side all right we
are in the process now so 1 by 2 and now
again south east south east south east
so 2 times 8 that is 16 all right and
minus 3 times 4 that's minus
and minus 1 times minus 2 that's plus 2
and then - we need to subtract now again
south west south west southwest so minus
3 times minus 2 you'll get plus 6 and
then 8 times 1 minus 1 that's why I
write here minus 1 so it will be minus 8
and 4 times 2 that's going to be 8 all
right and let me close it it's zero so
here 1 by 2 now 16 minus 12 that's 4 4
plus 2 that's 6 and then minus now 6
months 8 that's minus 2 or simply you
can cancel this both and it's going to
be 6 equals 0 so finally what we get
that 0 over 2 equals to 0 so further if
I go what I will get I will get 0 equals
to 0 right so left hand side is equal
with right hand side it means this shows
that there is there is not there is no
formation of there is no formation of
triangle there is no formation of
triangle there is no formation of
triangle it means it means those all
points are those all points are in a
line points are in a line all right
hence hence point ABC are collinear
point collinear point all right verified
alright catch in the next video with
next concept good bye