To determine the lengths of the two adjacent sides of the parallelogram, we need to solve for [tex]\( n \)[/tex] and then use it to find the specific side lengths.
1. Identify the expressions for the sides:
- One side of the parallelogram is [tex]\( 5n - 6 \)[/tex] cm.
- The opposite side is [tex]\( 3n - 2 \)[/tex] cm.
- An adjacent side is [tex]\( 2n + 3 \)[/tex] cm.
2. Set up an equation for the opposite sides being equal:
Since opposite sides of a parallelogram are equal, we set the expressions for the opposite sides equal to each other:
[tex]\[
5n - 6 = 3n - 2
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
[tex]\[
5n - 6 = 3n - 2
\][/tex]
Subtract [tex]\( 3n \)[/tex] from both sides:
[tex]\[
2n - 6 = -2
\][/tex]
Add 6 to both sides:
[tex]\[
2n = 4
\][/tex]
Divide by 2:
[tex]\[
n = 2
\][/tex]
4. Substitute [tex]\( n = 2 \)[/tex] into the expressions for the side lengths:
- Length of the side [tex]\( 5n - 6 \)[/tex]:
[tex]\[
5(2) - 6 = 10 - 6 = 4 \, \text{cm}
\][/tex]
- Length of the adjacent side [tex]\( 2n + 3 \)[/tex]:
[tex]\[
2(2) + 3 = 4 + 3 = 7 \, \text{cm}
\][/tex]
5. Conclusion:
The lengths of the two adjacent sides of the parallelogram are [tex]\( 4 \, \text{cm} \)[/tex] and [tex]\( 7 \, \text{cm} \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4 \, \text{cm} \text{ and } 7 \, \text{cm}} \][/tex]
