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A. Are not collinear

B. Cannot be plotted

C. Are not defined

D. Are collinear

Answer

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Where, $m = $ slope

$\left( {{x_1},{y_1}} \right) = $ Coordinate of first point in the line

$\left( {{x_2},{y_2}} \right) = $ Coordinate of second point in the line

The given points are $A\left( {{x_1},{y_1}} \right) = \left( {7,8} \right),B\left( {{x_2},{y_2}} \right) = \left( { - 5,2} \right)$ and $C\left( {{x_3},{y_3}} \right) = \left( {3,6} \right)$.

Now suppose three points $A\left( {{x_1},{y_2}} \right),B\left( {{x_2},{y_2}} \right)$ and $C\left( {{x_3},{y_3}} \right)$ are collinear, then slope of any two points be ${m_{AB}} = {m_{BC}} = {m_{AC}}$

Slope ${m_{AB}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{2 - 8}}{{ - 5 - 7}} = \dfrac{1}{2}$

Slope ${m_{BC}} = \dfrac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}} = \dfrac{{6 - 2}}{{3 + 5}} = \dfrac{1}{2}$ and

Slope ${m_{AC}} = \dfrac{{{y_3} - {y_1}}}{{{x_3} - {x_1}}} = \dfrac{{6 - 8}}{{3 - 7}} = \dfrac{1}{2}$

Therefore, slope of ${m_{AB}} = $Slope of ${m_{BC}} = $Slope of ${m_{AC}}$, that is slope of any two points are same.

Therefore, the given points $A,B$ and $C$ are collinear.

Let us plot the given points in a graph.

This showed that the points lie on the same line. So the given points are collinear.

If slope of $AB = $ slope of $BC = $ slope of $CA$, then $A$, $B$ and $C$ are collinear.

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