Prove that the points (−2, 5),

Question:

Prove that the points (−2, 5), (0, 1) and (2, −3)  are collinear.

Solution:

The distance $d$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula

$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}^{2}\right)}$

For three points to be collinear the sum of distances between two pairs of points should be equal to the third pair of points.

The given points are (25), (01) and (23)

Let us find the distances between the possible pairs of points.

$A B=\sqrt{(-2-0)^{2}+(5-1)^{2}}$

$=\sqrt{(-2)^{2}+(4)^{2}}$

$=\sqrt{4+16}$

$A B=2 \sqrt{5}$

$A C=\sqrt{(-2-2)^{2}+(5+3)^{2}}$

$=\sqrt{(-4)^{2}+(8)^{2}}$

$=\sqrt{16+64}$

$A C=4 \sqrt{5}$

$B C=\sqrt{(0-2)^{2}+(1+3)^{2}}$

$=\sqrt{(-2)^{2}+(4)^{2}}$

$=\sqrt{4+16}$

$B C=2 \sqrt{5}$

We see that $A B+B C=A C$

Since sum of distances between two pairs of points equals the distance between the third pair of points the three points must be collinear.

Hence we have proved that the three given points are.

 

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