Alright, let's find the lengths of two adjacent sides of the parallelogram for different values of [tex]\( n \)[/tex].
The lengths of the sides are given as:
- One side: [tex]\( 5n - 6 \)[/tex]
- Another side: [tex]\( 2n + 3 \)[/tex]
Let's calculate these lengths for some specific integer values of [tex]\( n \)[/tex].
### Case 1: [tex]\( n = 2 \)[/tex]
1. First side: [tex]\( 5(2) - 6 = 10 - 6 = 4 \)[/tex]
2. Adjacent side: [tex]\( 2(2) + 3 = 4 + 3 = 7 \)[/tex]
So for [tex]\( n = 2 \)[/tex], the lengths are [tex]\( 4 \)[/tex] cm and [tex]\( 7 \)[/tex] cm.
### Case 2: [tex]\( n = 4 \)[/tex]
1. First side: [tex]\( 5(4) - 6 = 20 - 6 = 14 \)[/tex]
2. Adjacent side: [tex]\( 2(4) + 3 = 8 + 3 = 11 \)[/tex]
So for [tex]\( n = 4 \)[/tex], the lengths are [tex]\( 14 \)[/tex] cm and [tex]\( 11 \)[/tex] cm.
### Case 3: [tex]\( n = 7 \)[/tex]
1. First side: [tex]\( 5(7) - 6 = 35 - 6 = 29 \)[/tex]
2. Adjacent side: [tex]\( 2(7) + 3 = 14 + 3 = 17 \)[/tex]
So for [tex]\( n = 7 \)[/tex], the lengths are [tex]\( 29 \)[/tex] cm and [tex]\( 17 \)[/tex] cm.
### Case 4: [tex]\( n = 13 \)[/tex]
1. First side: [tex]\( 5(13) - 6 = 65 - 6 = 59 \)[/tex]
2. Adjacent side: [tex]\( 2(13) + 3 = 26 + 3 = 29 \)[/tex]
So for [tex]\( n = 13 \)[/tex], the lengths are [tex]\( 59 \)[/tex] cm and [tex]\( 29 \)[/tex] cm.
Given these calculated lengths, we compare with the provided options:
- 4 cm and 7 cm for [tex]\( n = 2 \)[/tex]
- 14 cm and 11 cm for [tex]\( n = 4 \)[/tex]
- 29 cm and 17 cm for [tex]\( n = 7 \)[/tex]
- 59 cm and 29 cm for [tex]\( n = 13 \)[/tex]
Among the options:
- 2 cm and 2 cm: This is not possible with the given values.
- 4 cm and 7 cm: This corresponds to [tex]\( n = 2 \)[/tex].
- 7 cm and 9 cm: This is not a combination we have found.
- 13 cm and 19 cm: This is not a combination we have found.
Thus, the correct answer from the given options is:
[tex]\[ \boxed{4 \text{ cm and } 7 \text{ cm}} \][/tex]
