★ AIMS Think+ · Q7

Given:

The three points.

To Find:

Whether they are collinear.

Property:

The longest distance must equal the sum of the other two for collinearity.

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

Solution:

RB = $\sqrt{3^2+{\left(-4\right)}^2}$ = 5.

BC = $\sqrt{6^2+{\left(-7\right)}^2}$ = $\sqrt{85}$; RC = $\sqrt{9^2+{\left(-11\right)}^2}$ = $\sqrt{202}$.

If 5 + $\sqrt{85}$ = $\sqrt{202}$, squaring gives 110 + 10$\sqrt{85}$ = 202, hence $\sqrt{85}$ = 9.2.

But 9.$2^2$ = 84.64, not 85. Thus the required equality fails.

Direction check: doubling the horizontal change 3 → 6 would require vertical change −4 → −8, not −7.

Results:

The points are not collinear.