State whether the following statements are true or false. Justify your answer.
Points \( \mathrm{A}(3,1), \mathrm{B}(12,-2) \) and \( \mathrm{C}(0,2) \) cannot be the vertices of a triangle.
Given:
Points \( \mathrm{A}(3,1), \mathrm{B}(12,-2) \) and \( \mathrm{C}(0,2) \) cannot be the vertices of a triangle.
To do:
We have to find whether the given statement is true or false.
Solution:
We know that,
If the points \( \mathrm{A}(3,1), \mathrm{B}(12,-2) \) and \( \mathrm{C}(0,2) \) are collinear, then the area of triangle formed by the points is 0.
Area of a triangle $=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$
Therefore,
Area of the given triangle $=\frac{1}{2}[3(-2-2)+12(2-1)+0(1+2)]$
$=\frac{1}{2}[3(-4)+12(1)+0]$
$=\frac{1}{2}(-12+12)$
$=0$
The area of the triangle formed by the given points is 0.
Therefore, the points \( A(3,1),B(12,-2) \) and \( C(0,2) \) are collinear.
This implies,
Points \( \mathrm{A}(3,1), \mathrm{B}(12,-2) \) and \( \mathrm{C}(0,2) \) cannot be the vertices of a triangle.
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