Question

Determine whether the given points set of points are collinear or not.
(7,-2),(5,1),(3,4)

Answer 1

Points:

• $A(x_{1}, y_{1}) = (7,-2)$
• $B(x_{2}, y_{2}) = (5,1)$
• $C(x_{3}, y_{3}) = (3,4)$

If these points are Collinear then the triangle formed by joining these points must be of 0 units.

We know that,

$ar(\triangle ABC) = \lvert \frac{1}{2}[\: x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}) \: ]\; \rvert$

So,

$\Rightarrow \quad ar(\triangle ABC) = 0$

$\Rightarrow \quad \lvert \frac{1}{2}[\: 7(1 - 4) + 5(4 - (-2)) + 3(-2 - 1) \: ] \rvert = 0 \\ \\ \Rightarrow \quad ( \: 7 \cdot -3 + 5 \cdot 6 + 3 \cdot -3 \: ) = 0 \\ \Rightarrow \quad -21 + 30 - 9 = 0 \\ \Rightarrow \quad \cancel{-30} \cancel{+ 30 } = 0$

Since, the area is equal to 0 units, Therefore the points are Collinear.