To determine the lengths of the two adjacent sides of the parallelogram, we need to focus on the given information and follow these logical steps:
1. Identify the Equations for Opposite Sides:
We know two opposite sides of the parallelogram have lengths [tex]\(5n - 6 \, \text{cm}\)[/tex] and [tex]\(3n - 2 \, \text{cm}\)[/tex].
2. Use the Property of Parallelograms:
In a parallelogram, opposite sides are equal. Therefore, we can set these two expressions equal to each other to find the value of [tex]\(n\)[/tex]:
[tex]\[
5n - 6 = 3n - 2
\][/tex]
3. Solve for [tex]\(n\)[/tex]:
Rearrange the equation to solve for [tex]\(n\)[/tex]:
[tex]\[
5n - 3n = -2 + 6
\][/tex]
[tex]\[
2n = 4
\][/tex]
[tex]\[
n = 2
\][/tex]
4. Calculate the Lengths of the Sides:
Substitute [tex]\(n = 2\)[/tex] back into the expressions for the sides to find their lengths.
- For the side given by [tex]\(5n - 6\)[/tex]:
[tex]\[
5(2) - 6 = 10 - 6 = 4 \, \text{cm}
\][/tex]
- For the side given by [tex]\(2n + 3\)[/tex]:
[tex]\[
2(2) + 3 = 4 + 3 = 7 \, \text{cm}
\][/tex]
5. Conclude the Lengths:
Therefore, the lengths of the two adjacent sides of the parallelogram are [tex]\(4 \, \text{cm}\)[/tex] and [tex]\(7 \, \text{cm}\)[/tex].
The correct answer is:
[tex]\[
\boxed{4 \, \text{cm} \text{ and } 7 \, \text{cm}}
\][/tex]
