I've been stuck on this for a couple of days now.

  see the attachment

Step-by-step explanation:

The attached array shows the column number to which each tile needs to be dragged. The table has the same layout as the tile array, so matching tiles to column numbers should be straightforward.

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There are a couple of tricks that make this fairly simple to do. The square roots will only be associated with lengths, so anything that says "length" will go into one of the square root columns.

The square roots are multiples of √10 and of √5. It helps to recognize that ...

so, if the components of the difference of endpoints are in the ratio 1:3, they will go in the √10 column. If they are in the ratio 1:2, they will go in the √5 column.

Consider the upper left tile. Endpoints of the segment are (0, -1) and (4, 7). Their differences are (4, 8), numbers that have the ratio 1:2. This tile goes in column 4, under 4√5.

Consider the tile below that one, the first tile in the second row. Endpoints of that segment are (2, -1) and (4, 5). The differences are (2, 6), numbers that have the ratio 1:3. This tile goes in column 3, under 2√10.

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The "midpoint" tiles are more easily classified. The midpoint is the average of the endpoint coordinates, so is half their sum. If the sum of the x-coordinates is 4, then the midpoint will have an x-coordinate of 2, and the tile will go in the first column.

If the sum of x-coordinates is 6, the midpoint will have an x-coordinate of 3, and the tile will go in the second column. All you have to do is add the x-coordinates to find the column the tile goes to.

step-by-step explanation: