In the $3 \times 8$ array, the dots are evenly spaced horizontally and vertically with each dot 1 cm from the nearest neighboring dots. In simplest radical form, what is the number of units in the length of the longest segment containing exactly 3 dots, two of which are the segment's endpoints?
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Any thoughts? What's the longest segment you've produced so far? – lulu Apr 13 '19 at 23:23
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I found the segment sqrt(10)+ sqrt(17), but then I found that it was wrong. I found this by just drawing a line from the top left corner to the middle dot, and then another segment to the bottom right corner. – Awxx178 Apr 13 '19 at 23:26
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Well, I expect they mean a straight line segment. Draw the picture and try to make long line segments. There really aren't a lot of options. – lulu Apr 13 '19 at 23:30
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How are you sure if a segment is straight if it passes through 3 points? – Awxx178 Apr 13 '19 at 23:33
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Draw the grid on the usual $(x,y)-$ plane. Corners at $(0,0),(2,0),(2,7), (0,7)$. Now you know what straight lines are (constant slope). For instance the segment connecting $(0,0), (1,1), (2,2)$ is a straight line (part of $y=x$). Probably not maximal though. – lulu Apr 13 '19 at 23:35
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Does the below figure help at all?
David G. Stork
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1Oh! Ok, thank you so much! I guess since there is no "middle" point for the 38 rectangle, the only way to find the longest segment is using a 37 rectangle. – Awxx178 Apr 13 '19 at 23:40
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