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.

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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Updated on: 10-Oct-2022

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