Question

determine the distance formula whether the points (4, 7)( - 1, 2 )and (2, 5 )are collinear

Answer 1

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Answer 2

GIven points are (4,7), (-1,2) and (2,5).

Let's given them some name like A, B , C to easily identify the points.

So now we have points A(4,7), B(-1,2) and C(2,5)

Now we can apply distance formula to find distance between two points (x1,y1) and (x2,y2).

$distance =\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Lets start with points A(4,7), B(-1,2)

$AB=\sqrt{(-1-4)^2+(2-7)^2}$

$AB=\sqrt{(-5)^2+(-5)^2}$

$AB=\sqrt{25+25}$

$AB=\sqrt{50}$

$AB=5\sqrt{2}$...(i)

Now find distance between B(-1,2) and C(2,5)

$BC=\sqrt{(2--1)^2+(5-2)^2}$

$BC=\sqrt{(3)^2+(3)^2}$

$BC=\sqrt{9+9}$

$BC=\sqrt{18}$

$BC=3\sqrt{2}$...(ii)

Now find distance between A(4,7) and C(2,5)

$AC=\sqrt{(2-4)^2+(5-7)^2}$

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

$AC=\sqrt{4+4}$

$AC=\sqrt{8}$

$AC=2\sqrt{2}$...(iii)

Now if we add value of BC and AC then we get:

BC+AC=3\sqrt{2} [/tex]+2\sqrt{2} [/tex]=5\sqrt{2} [/tex]

Which is same as the value of AB

so we are getting sum two lines equal to the third line

BC+AC=AB

Hence given points are collinear.