Adams, Greensburg, Middletown, and Waldron are four towns in Indiana on a nearly straight highway. If the highway is represented by a line, the towns can be represented by four points: A, G, M, and W. Given that M is the midpoint of AW and MG=MA+GA, determine the proper order of the four towns.

Given:

Points A, G, M, and W on a line represent the four towns Adams, Greensburg, Middletown, and Waldron respectively.

M is the midpoint of AW and MG=MA+GA.

To find:

The proper order of the four towns.

Solution:

It is given that M is the midpoint of AW. So, point M is between points A and W.

Order of these three points is A, M, W.           ... (i)

Now, MG=MA+GA. It is possible only if the point A lies between M and G because of segment addition property.

Order of these three points is G, A, M.           ... (ii)

Using (i) and (ii), we get

Order of four towns = G, A, M, W

Therefore, the order of four towns is Greensburg, Adams, Middletown, and Waldron.

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