★ AIMS Think+ · Q6

Given:

The three specified points.

To Find:

Whether they lie on one straight line.

Property:

For distinct collinear points with A between M and G, MA + AG = MG.

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

Solution:

MA = $\sqrt{3^2+4^2}$ = 5.

AG = $\sqrt{6^2+8^2}$ = 10.

MG = $\sqrt{9^2+{12}^2}$ = 15.

MA + AG = 5 + 10 = 15 = MG. Equality in the triangle inequality places A on MG.

Results:

The points are collinear; A is between M and G.