Question

determine whether points are collinear (-2,2) (0, 4) and (2,-10)

Answer 1

Given ,

The three point are A(-2,2) , B(0, 4) and C(2,-10)

$\underline{ \sf{First \: method :}}$

We know that , if area of Δ ABC is zero , then three points A , B and C are collinear

$\boxed{ \sf{Area \: of \: triangle = \frac{1}{2} | x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2})| }}$

Thus ,

$\sf \mapsto Area = \frac{1}{2} |2 \{ 4 - ( - 10)\} + 0 \{ - 10 - 2 \} + 2 \{ - 2 - 4 \}| \\ \\ \sf \mapsto Area = \frac{1}{2} |2 \{14 \} + 0 + 2 \{ 6\}| \\ \\ \sf \mapsto Area = \frac{1}{2} |28 - 12| \\ \\ \sf \mapsto Area = \frac{16}{2} \\ \\ \sf \mapsto Area =8 \: \: {units}^{2}$

Here , the area of Δ ABC ≠ 0

✭ The given three points are collinear

$\underline{ \sf{Second \: method :}}$

We know that ,

If three points A , B and C are collinear , then the slope of AB = slope of BC

$\boxed{ \sf{Slope \: (m) = \frac{ y_{2} - y_{1} }{x_{2} - x_{1} } }}$

Thus ,

Slope of AB = {4 - 2}/{0 - (-2)}

Slope of AB = 2/2

Slope of AB = 1

Similarly , the slope of BC will be

Slope of BC = {-10 - 4}/}2 - 0}

Slope of BC = -14/2

Slope of BC = -7

Here , Slope of AB ≠ Slope of BC

✭ The given three points are collinear