Question

quadrilateral abde is a rectangle. ab = 10 cm and ae = 16 cm.
through which two points could a line of rotation be placed so that the base of the resulting cylinder will have a radius of 5 cm?
○ d and e
○ b and d
○ c and f
○ g and h

Explanation:

Step1: Recall cylinder formation by rotation

When a rectangle is rotated about a line (axis of rotation), the side perpendicular to the axis becomes the radius of the base of the cylinder. The length of the side parallel to the axis becomes the height. Given the radius of the base of the cylinder is \( 5\space\text{cm} \), we need to find the axis such that the perpendicular side to it has length \( 2\times5 = 10\space\text{cm} \) (diameter) or the side itself is \( 5\space\text{cm} \) (radius). Wait, actually, when rotating a rectangle around an axis, the radius is the distance from the axis to the opposite side.

Given \( AB = 10\space\text{cm} \) and \( ABDE \) is a rectangle. Let's analyze each option:

- Option 1: D and E. The line DE is horizontal (from the diagram). Rotating around DE, the distance from DE to AB (or AE?) Wait, no. Let's look at the diagram. ABDE is a rectangle, so AB is vertical (length 10 cm), AE is horizontal (length 16 cm). Wait, maybe AB is vertical with length 10 cm, so AB is from A (bottom) to B (top), length 10 cm. AE is from A (right) to E (left), length 16 cm. So the rectangle has length AE = 16 cm (horizontal) and height AB = 10 cm (vertical).

Now, for the radius of the cylinder to be 5 cm, the axis of rotation should be such that the distance from the axis to the opposite side is 5 cm, or the side being rotated has length related to the radius. Wait, when you rotate a rectangle around a vertical axis (parallel to AB), the radius would be the horizontal distance (AE direction). Wait, maybe better to think: if we rotate around a horizontal line (parallel to AE), then the radius would be half of AB (since AB is vertical, length 10 cm, so half is 5 cm). So the axis should be a horizontal line (parallel to AE) such that the distance from the axis to AB (or DE) is 5 cm. Looking at the diagram, points G and H: G is on AB, H is on DE, and the marks suggest they are midpoints? Wait, AB is 10 cm, so if we rotate around GH (the line through G and H), which is horizontal? Wait, no, G and H are vertical? Wait, the diagram: D---C---B (top horizontal), E---F---A (bottom horizontal), D to H to E (left vertical), B to G to A (right vertical). So GH is the vertical line through G (on AB) and H (on DE). Wait, no, maybe G and H are on the vertical sides. Wait, the options: G and H. Let's check:

If we rotate the rectangle ABDE around the line GH (which is vertical, passing through G and H), then the horizontal distance from GH to AE (or BD) would be... Wait, maybe I got the axes wrong. Let's re-express:

AB is vertical (length 10 cm), so AB is from A (y=0) to B (y=10). AE is horizontal (length 16 cm), from A (x=0) to E (x=-16). So the rectangle has corners at A(0,0), B(0,10), D(-16,10), E(-16,0).

Now, rotating around a vertical line (parallel to AB, i.e., parallel to y-axis) would create a cylinder with radius equal to the horizontal distance from the axis to the sides AE or BD. Wait, no, if we rotate around the line x = -8 (midpoint of AE, since AE is 16 cm long, from x=0 to x=-16, midpoint at x=-8), but that's not an option. Wait, the other option: rotating around a horizontal line (parallel to AE, i.e., parallel to x-axis) would create a cylinder with radius equal to the vertical distance from the axis to the top or bottom. Since AB is 10 cm (vertical), the midpoint of AB is at y=5, so a horizontal line at y=5 would be the axis, and rotating around it would give a radius of 5 cm (since the distance from y=5 to A (y=0) or B (y=10) is 5 cm). But the options are D and E, B and D, C and F, G and H.

Wait, let's look at the marks on the diagram: AB has two marks (equal segments), so G is the midpoint of AB (since AB is 10 cm, midpoint at 5 cm from A and B). Similarly, H is the midpoint of DE. So the line GH is vertical (connecting midpoints of AB and DE). Wait, no, GH is vertical? Wait, AB is vertical, DE is vertical? No, ABDE is a rectangle, so AB and DE are vertical sides (length 10 cm), AE and BD are horizontal sides (length 16 cm). So AB is from A (bottom) to B (top, vertical), DE is from D (top) to E (bottom, vertical). So the vertical sides are AB (right) and DE (left), horizontal sides are BD (top) and AE (bottom). So points G is on AB (right vertical), H is on DE (left vertical). So the line GH is vertical, connecting midpoints of AB and DE. Now, if we rotate the rectangle around the vertical line GH, what happens? The horizontal sides (BD and AE) are length 16 cm, so rotating around GH (vertical axis) would create a cylinder with radius equal to the horizontal distance from GH to AE (or BD). Wait, AE is from A (0,0) to E (-16,0), BD is from B (0,10) to D (-16,10). The line GH is at x=0? No, wait, AB is at x=0 (right side), DE is at x=-16 (left side). So midpoint of AB is at x=0, y=5 (G), midpoint of DE is at x=-16, y=5 (H). So the line GH is vertical, from (0,5) to (-16,5)? No, that can't be. Wait, maybe the diagram is different: D---C---B (top horizontal, length equal to AE=16 cm), E---F---A (bottom horizontal, length 16 cm), D to H to E (left vertical, length AB=10 cm), B to G to A (right vertical, length AB=10 cm). So G is on AB (right vertical), H is on DE (left vertical), and F is on AE (bottom horizontal), C is on BD (top horizontal).

Now, the radius of the cylinder is 5 cm. So when we rotate the rectangle around an axis, the radius is the distance from the axis to the side being rotated. Let's check each option:

- Option D and E: Line DE is the left vertical side (length 10 cm). Rotating around DE, the radius would be the horizontal distance from DE to AB, which is the length of AE (16 cm)? No, that can't be. Wait, no, if we rotate the rectangle around DE (vertical line), then the side AE (horizontal, length 16 cm) would rotate, creating a cylinder with radius equal to the length of AE? No, that doesn't make sense. Wait, maybe I have the axes reversed. Let's think again: when you rotate a rectangle around a vertical axis (parallel to AB), the radius is the horizontal length (AE or BD), and the height is the vertical length (AB). But we need radius 5 cm, so the horizontal length should be 5 cm? No, AE is 16 cm. Wait, maybe the rectangle is being rotated around a horizontal axis (parallel to AE), so the radius is the vertical length. AB is 10 cm, so if we rotate around a horizontal axis that is 5 cm from A and B (i.e., the midline of AB), then the radius would be 5 cm. Looking at the diagram, the line through G and H: G is on AB, H is on DE, and since AB and DE are vertical (length 10 cm), the midpoint of AB is at 5 cm from A (so G is at 5 cm from A), and midpoint of DE is at 5 cm from E (so H is at 5 cm from E). So the line GH is horizontal? Wait, no, in the diagram, D---C---B (top horizontal), E---F---A (bottom horizontal), D to H to E (left vertical), B to G to A (right vertical). So H is on DE (left vertical), G is on AB (right vertical), so GH is a horizontal line connecting H (on DE) and G (on AB), at the midpoint (5 cm from A and B, 5 cm from D and E). So rotating the rectangle around GH (the horizontal line at y=5) would result in a cylinder where the radius is the distance from GH to A (or B), which is 5 cm (since AB is 10 cm, from y=0 to y=10, so midpoint at y=5, distance from y=5 to y=0 is 5 cm). Therefore, the axis of rotation is GH, i.e., points G and H.

Let's verify other options:

- D and E: Line DE is horizontal (top horizontal side). Rotating around DE, the radius would be the distance from DE to AE (bottom horizontal side), which is AB = 10 cm, so radius 10 cm, not 5 cm.

- B and D: Line BD is horizontal (top horizontal side, same as DE? No, BD is from B to D, which is horizontal, length 16 cm. Rotating around BD, radius would be AB = 10 cm, same as above.

- C and F: Line CF is horizontal, midpoint? But CF is on the top and bottom horizontal sides? Not sure, but likely not the midline for radius 5 cm.

- G and H: As explained, rotating around GH (midline of AB and DE) gives radius 5 cm (distance from GH to A or B is 5 cm).

Step2: Confirm the correct option

So the correct option is G and H.

Answer:

D. G and H