is a directed segment with endpoints A(2,2) and B(14,14). In two or more complete sentences, explain how you would verify that Point X, with a coordinate of (6,6), divides in a 1:2 ratio?

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Explanation:

The length of a line segment is computed from the differences between corresponding end-point coordinates. That is, one needs to know the difference of x-coordinates and the difference of y-coordinates. If the point X is on line segment AB and divides it accordingly, then we have the relation ...

  (X - A) : (B - X) = 1 : 2

This will be true for both the x-coordinates and the y-coordinates of the respective differences. Here, that relation evaluates to ...

  (X -A) = (6, 6) -(2, 2) = (4, 4)

  (B -X) = (14, 14) -(6, 6) = (8, 8)

Then the ratios of coordinates are ...

  x-coordinates: 4 : 8 = 1 : 2

  y-coordinates: 4 : 8 = 1 : 2

These are the same ratios, so we can conclude that X is on segment AB and divides it in the required ratio.