Question

State wheather the following points are collinear or non collinear A(1,3) ,B(3,-2),C(-3,16)​

Answer 1

${\large{\underbrace{\underline{\bf{Required\:Answer:-}}}}}$

$\underline{\underline{\frak{\red\dag\;Given\;Points:}}}$

• A = (1 , 3)
• B = (3 , -2)
• C = (-3 , 16)

$\underline{\underline{\frak{\red\dag\;Solution:}}}$

» We are supposed to prove that these poins are collinear.

• Collinear Points : The points which line on a straight line.

» We know that, if the points are collinear then, they would be forming a line, if connected.

» So, If these points are on a line, they won't be forming any figure , so, we can say that the area of a triangle formed by these points would be 0 as, the points are collinear. If the area is not 0 then the points would not be collinear.

Compare the points A, B & C with $\rm (x_1,y_1); (x_2,y_2)\:\:and\:\:(x_3,y_3)$

So,

• $\rm x_1,y_1=1,3$
• $\rm x_2,y_2=3,-2$
• $\rm x_3,y_3=-3,16$

Thus,

• $\rm x_1=1 , y_1=3$
• $\rm x_2=3,y_2=-2$
• $\rm x_3=-3,y_3=16$

Put these values in the formula for finding the area of a triangle.

$\odot \: \boxed{ \sf \: Area_{ \: triangle} = \frac{1}{2} |x _{1} (y _{2} - y _{3}) + x _{2}(y _{3} - y _{1}) + x _{3}( y _{1} - y _{2}) | }$

Substitute the values in the formula..

$\colon \rarr \sf \: Area = \frac{1}{2} |1( - 2 - 16) + 3(16 - 3) +( - 3)(3 - ( - 2))| \\ \\ \colon \rarr \sf \: Area = \frac{1}{2} |1( - 18) + 3(13) - 3(3 + 2)| \\ \\ \colon \rarr \sf \: Area \: = \frac{1}{2} | - 18 + 39 - 15| \\ \\ \colon \sf\rarr \: Area \: = \frac{1}{ \cancel2} | \cancel6| \\ \\ \colon \implies \: \underline{ \boxed{ \frak{ \pink{Area = 3 \: {unit}^{2} }}}}$

As you can see :

$\quad \quad \quad \quad\colon \dashrightarrow \: \boxed{ \purple{\bf \: Area \ne\red{0}}}$

So,

$\therefore\sf{\underline{Points\;A,B\;and\;C\;are\;\bf{non\;collinear.}}}$