rotating a rectangle
through which points should a line of rotation be placed to create a cylinder with a radius of 3 units?
e and j
f and i
j and g
g and e

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# Brief Explanations:
To create a cylinder with radius 3 units, the line of rotation should be parallel to a side of length 4 (since radius is half the diameter, and we need a distance of 3 from the line of rotation to the side being rotated). Looking at the rectangle, the line through E and J (vertical side) would have the horizontal sides (length 3 from E to F and F to G, etc.) at a distance of 3? Wait, no. Wait, when rotating a rectangle around a line, the radius is the distance from the line of rotation to the side perpendicular to the axis. Wait, the rectangle has sides 4 (vertical) and 6 (horizontal, 3+3). Wait, no, the horizontal segments are 3 (E to F) and 3 (F to G), so total horizontal length 6? No, the vertical sides are 4 (E to J and G to H). Wait, the options: E and J is a vertical line. If we rotate the rectangle around EJ, then the horizontal sides (like FG or EH? Wait, no, the rectangle has vertices E, F, G, H, I, J? Wait, the diagram: E---F---G, with E to F 3, F to G 3; G---H (vertical 4), H---I---J (horizontal, I is midpoint? H to I to J, with H to I? Wait, no, the vertical sides: E to J (4 units), G to H (4 units). So the rectangle is E(0,4), F(3,4), G(6,4), H(6,0), I(3,0), J(0,0). Wait, no, maybe E(0,4), F(3,4), G(6,4), H(6,0), I(3,0), J(0,0). Then the vertical sides are EJ (x=0, y from 0 to 4) and GH (x=6, y from 0 to 4). The horizontal sides are EG (y=4, x from 0 to 6) and JH (y=0, x from 0 to 6), and FG (x from 3 to 6, y=4), EJ (x=0, y from 0 to 4), etc. Wait, the radius needed is 3. So when rotating around a vertical line, the horizontal distance from the line to the opposite side should be 3. Wait, EJ is at x=0. The side FG is at x=3 (from F(3,4) to G(6,4)? No, F is at 3 from E, so E is (0,4), F(3,4), G(6,4), H(6,0), I(3,0), J(0,0). So the horizontal distance from EJ (x=0) to FG (x=3) is 3? Wait, no, FG is from x=3 to x=6. Wait, maybe I got the diagram wrong. Alternatively, the line through E and J: E is (0,4), J is (0,0). So the line EJ is the left vertical side. The horizontal sides: from E(0,4) to F(3,4) is 3 units, so the distance from EJ (x=0) to EF (x from 0 to 3) is 0 to 3? No, wait, when rotating the rectangle around EJ, the side opposite to EJ would be GH? No, GH is at x=6. Wait, no, maybe the rectangle is E, F, G, H, with E to F 3, F to G 3? No, that would be E to G 6. Wait, the vertical sides are E to J (4) and G to H (4). So the rectangle is E(0,4), F(3,4), G(6,4), H(6,0), I(3,0), J(0,0). So if we rotate around EJ (x=0), then the horizontal distance from EJ to FG (x=3 to 6) is 3 (from x=0 to x=3 is 3 units). Wait, no, FG is from x=3 to x=6, so the distance from EJ (x=0) to FG (x=3) is 3. So when rotating the rectangle around EJ, the side FG (or the horizontal segment) would be at distance 3 from EJ, so the radius would be 3. Wait, but let's check the options. The options are E and J, F and I, J and G, G and E. F and I: F is (3,4), I is (3,0) – that's a vertical line at x=3. Rotating around FI would have the sides EJ (x=0, distance 3 from x=3) and GH (x=6, distance 3 from x=3). So radius 3. Wait, that makes sense. Wait, maybe I messed up. Wait, the rectangle has width 6 (3+3) and height 4. If we rotate around the vertical line through F and I (x=3), then the distance from x=3 to x=0 (EJ) is 3, and to x=6 (GH) is 3. So rotating the rectangle around FI (x=3) would create a cylinder with radius 3 (since the horizontal sides are 3 units from the axis). Wait, but the question is which line of rotation (through which points) gives radius 3. Let's re-examine the options:

- E and J: vertical line at x=0. The distance from x=0 to x=3 (F) is 3, but the other side is x=6 (G), distance 6. So rotating around EJ would have radius 3 (from EJ to EF) but the other side would be radius 6? No, that can't be. Wait, no, the rectangle is E-F-G-H-I-J-E? Wait, maybe the rectangle is E, F, G, H, with E to F 3, F to G 3, G to H 4, H to I 3, I to J 3, J to E 4? No, the diagram shows E---F---G (3,3), G---H (4), H---I---J (3,3), J---E (4). So it's a rectangle with length 6 (3+3) and height 4. So the vertical sides are EJ (4) and GH (4), horizontal sides EH? No, E to F to G to H to I to J to E. So it's a rectangle with length 6 (E to G is 6) and height 4 (E to J is 4). Now, when rotating a rectangle around a vertical line (parallel to the height), the radius is the horizontal distance from the line to the horizontal sides. So to get radius 3, the line should be 3 units from the horizontal sides. The horizontal sides are at x=0 (EJ) and x=6 (GH)? No, E is at (0,4), F at (3,4), G at (6,4), H at (6,0), I at (3,0), J at (0,0). So the horizontal midline is at x=3 (through F and I). So the line through F and I (x=3) is the vertical line at x=3, which is 3 units from x=0 (EJ) and 3 units from x=6 (GH). So rotating around FI (x=3) would have radius 3 (distance from FI to EJ is 3, FI to GH is 3). So the line through F and I (points F and I) would be the axis, and rotating the rectangle around it would create a cylinder with radius 3 (since the horizontal sides are 3 units from the axis, so when rotated, the radius is 3). Wait, but the options are E and J (x=0), F and I (x=3), J and G (diagonal? No, J and G: J is (0,0), G is (6,4) – that's a diagonal, not vertical or horizontal. G and E: G(6,4) and E(0,4) – horizontal line at y=4. Rotating around GE would create a cylinder with radius 4 (height 6), no. J and G: diagonal, radius would be variable. So the correct answer should be F and I. Wait, but let's check the options again. The options are:

- E and J: vertical line at x=0. Distance from x=0 to x=3 (F) is 3, but the other side is x=6 (G), distance 6. So rotating around EJ would have radius 3 (from EJ to EF) but the other side would be radius 6? No, that's not a cylinder. Wait, no, the rectangle is E-F-G-H, with E-F 3, F-G 3, G-H 4, H-I 3, I-J 3, J-E 4. So it's a rectangle with length 6 (E to G) and height 4 (E to J). So when rotating around EJ (vertical line), the side FG (length 3) is at x=3, so distance from EJ (x=0) to FG (x=3) is 3. So rotating FG around EJ would create a circle with radius 3. Wait, maybe the rectangle is being rotated such that the side FG or EH is rotated. Wait, I think I made a mistake earlier. Let's think: the radius of the cylinder is 3, so the diameter is 6? No, radius 3 means diameter 6. Wait, the rectangle has a side of length 6 (3+3). So if we rotate the rectangle around a vertical line that is 3 units from the side of length 6, then the radius would be 3. Wait, the line through E and J is vertical, and the horizontal side (from E to F to G) is length 6 (3+3). Wait, no, E to F is 3, F to G is 3, so E to G is 6. So the horizontal length is 6, vertical is 4. If we rotate the rectangle around the vertical line through E and J (x=0), then the side G-H-I-J? No, the side opposite to EJ is G-H, which is at x=6, distance 6 from EJ. So radius would be 6. But we need radius 3. So the line should be at x=3, which is through F and I (since F is (3,4) and I is (3,0)). So rotating around FI (x=3) would have the sides EJ (x=0, distance 3 from x=3) and GH (x=6, distance 3 from x=3). So the radius is 3, and the height is 4 (the vertical side). So that makes sense. Therefore, the correct option is F and I.
# Answer:
B. F and I

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