Answer :
Let's analyze the given rectangle [tex]\( ABDE \)[/tex] with dimensions [tex]\( AB = 10 \)[/tex] cm and [tex]\( AE = 16 \)[/tex] cm.
The question requires us to identify which pairs of points could be used as the line of rotation to form a cylinder with a radius of [tex]\( 5 \)[/tex] cm.
To solve this, let's consider the possible scenarios:
1. Rotating around [tex]\( AB \)[/tex]:
- If we rotate the rectangle around the side [tex]\( AB \)[/tex], then the length [tex]\( AE = 16 \)[/tex] cm would become the diameter of the cylinder's base.
- Since the diameter of the cylinder is twice the radius, the radius [tex]\( r \)[/tex] would be [tex]\( \frac{AE}{2} = \frac{16}{2} = 8 \)[/tex] cm.
- However, we need a radius of [tex]\( 5 \)[/tex] cm, so this scenario does not work.
2. Rotating around [tex]\( AE \)[/tex]:
- If we rotate the rectangle around the side [tex]\( AE \)[/tex], then the length [tex]\( AB = 10 \)[/tex] cm would become the diameter of the cylinder's base.
- The diameter of the cylinder would then be [tex]\( AB \)[/tex], making the radius [tex]\( r \)[/tex] [tex]\( \frac{AB}{2} = \frac{10}{2} = 5 \)[/tex] cm, which matches our required radius.
Therefore, the correct pair of points must lie along the side [tex]\( AE \)[/tex] (that would form the desired radius when rotated). Considering the sides more specifically:
- Points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] are such that rotating around this line through these points achieves the correct base radius of [tex]\( 5 \)[/tex] cm.
Thus, the line of rotation, ensuring the correct radius, passes through points [tex]\( B \)[/tex] and [tex]\( D \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
The question requires us to identify which pairs of points could be used as the line of rotation to form a cylinder with a radius of [tex]\( 5 \)[/tex] cm.
To solve this, let's consider the possible scenarios:
1. Rotating around [tex]\( AB \)[/tex]:
- If we rotate the rectangle around the side [tex]\( AB \)[/tex], then the length [tex]\( AE = 16 \)[/tex] cm would become the diameter of the cylinder's base.
- Since the diameter of the cylinder is twice the radius, the radius [tex]\( r \)[/tex] would be [tex]\( \frac{AE}{2} = \frac{16}{2} = 8 \)[/tex] cm.
- However, we need a radius of [tex]\( 5 \)[/tex] cm, so this scenario does not work.
2. Rotating around [tex]\( AE \)[/tex]:
- If we rotate the rectangle around the side [tex]\( AE \)[/tex], then the length [tex]\( AB = 10 \)[/tex] cm would become the diameter of the cylinder's base.
- The diameter of the cylinder would then be [tex]\( AB \)[/tex], making the radius [tex]\( r \)[/tex] [tex]\( \frac{AB}{2} = \frac{10}{2} = 5 \)[/tex] cm, which matches our required radius.
Therefore, the correct pair of points must lie along the side [tex]\( AE \)[/tex] (that would form the desired radius when rotated). Considering the sides more specifically:
- Points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] are such that rotating around this line through these points achieves the correct base radius of [tex]\( 5 \)[/tex] cm.
Thus, the line of rotation, ensuring the correct radius, passes through points [tex]\( B \)[/tex] and [tex]\( D \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
