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There are three types of sides, vertical, horizontal and diagonal. Do you need to used the Distance Formula to measures one or more of these types?
No.
Explanation: See solution.
We can find a figure's perimeter by adding the lengths of all of it's sides. For the school crossing, we can find the lengths of the vertical and horizontal sides without using the Distance Formula. Note that we assume that 1 unit on the coordinate plane is the same as 1 cm.
To measure the diagonal parts, we cannot do the same thing even though the top of the school crossing has coordinates that are integers as well. Let's try measuring the distance using a straightedge.
The measure of the diagonal part is about 2.8 and if we accepted rounded answers, we would be done. However, assuming that we want exact measurements, we have to use the Distance Formula. Therefore, our friend is not correct.