The regular 3 by 3 grid of dots above consists of evenly spaced rows

Let the rows be 1, 2 and 3 and columns be A, B and C. Thus the dots are A1, A2, A3, B1, B2, B3, C1, C2 and C3.
Total line segments = line segments having 3 dots + line segments having only 2 dots.

Since line segments can have 2 dots, total line segments = 9C2 = 36 (including line segments with 3 dots)
Now, we can count line segments with 3 dots also. There are 8 of them as follows:
A1A2A3
B1B2B3
C1C2C3
A1B1C1
A2B2C2
A3B3C3
A1B2C3
A3B2C1

Total line segments with 2 dots = 36 - 8 = 28.

Note: The line segments with 3 dots can be counted as two line segments with 2 dots.
Visually, required possibilities are:
- Horizontal: A1B1, B1C1, A2B2, B2C2, A3B3, B3C3
- Vertical: A1A2, A2A3, B1B2, B2B3, C1C2, C2C3
- 45° downhill: A1B2, B2C3, B1C2, A2B3
- <45° downhill: A1C2, A2C3
- >45° downhill: A1B3, B1C3
- 45° uphill: A3B2, B2C1, A2B1, B3C2
- <45° uphill: A2C1, A3C2
- >45° uphill: A3B1, B3C1

Answer E.

IMO E

I tried to count like in topLHS square: 6 lines possible
top RHS :5
bottom LHS: 5
bottom RHS: 4
then two lines in top as diagonals and two in bottom
then two lines from top extreme dots to center dot bottom
and similarly two lines from bottom extreme dots to center top.

Total: 28

Total is 24, there are 5 lines from the corners, times 4 is 24. Then there are 4 lines from the middle dot to the sides. So that is 24. If you take all lines from the centre you would get 28 but that counts the lines between middle and corners double

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the only solution for these sort of problems is to count all possible cases and guess (I guessed the answer E after counting 27 unique value with this method )

first I took 4 dots forming a square(bottom left) to form a line 4C2 = 6 lines , you also have a square on top right (+6)

then you have (from the top left point and bottorm right point ) 3 *2 new distinct lines
also middle point right with bottom center, and the parallel line to this line (according to the center of gravity of the whole square) +2

you have also 4 rectangles (top left point , middle top point , bottom left point , bottom middle point) you have 8

6+6+3*2+2+8=28 , E

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