Trying To Avoid Other Points

Relevant wiki: Distance between Two Points

I graphed few of the long segments that join two points and we see the longest one is the one joining the two corners; either $BC$ or $AD$ , which each has a length of $\sqrt{8^{2}+6^{2}} = \sqrt{100}=10$

First, notice that the line must be the diagonal of a rectangle, as any horizontal or vertical line, or any diagonal of a square will pass through at least one other point. The logical conclusion then would be that the diagonal of the 3X4 rectangle is the answer, but $\sqrt{64+16}=10$ is not one of the answer choices, so the next largest rectangle (3X2) must be the answer. Thus, our answer is $\sqrt{36+16}=\boxed{\sqrt{52}}$ .

Pythagorean Theorem! a=6 (Left or Right Side) b=8 (Top or Bottom Side) and * c=longest line segment *, diagonally.

8^2+6^2=c^2.

64+36=c^2 SO c=sqrt(100)

Sorry abou tthe sloppiness, I'm new =D

Actually you can connect two opposite corner points without crossing any point in between!

This can be shown by following : The slope of a line equals the change in height divided by the change in length, better known as $m = \frac{y_2-y_1}{x_2-x_1}$

If we imagine the dots as points on a grid with coordinates from bottom-left $(0,0)$ to upper-right $(4,3)$ using a usual coordinate system and drawing a line through these corner points, then $m =\frac{3-0}{4-0} =\frac{3}{4}$

All dots represent points with integers, therefore a point on this line has to have an $x$ ,which is 1, 2 or 3 to let the point be on the line between 0 and 4, and a $y = 1 + \frac {3}{4}x$ , which has to be an integer as well.

Only 0 and 4 would fullfill these requirements (or any other integer multiple of 4) and so no dot crosses the line !

You thought of a connection between $(0,0)$ and $(3,2)$ , which leads to $\sqrt{6^2+4^2}=\sqrt{52}$

But the correct answer is $\boxed{\sqrt{8^2+6^2}=\sqrt{100} = 10}$