Juanita is cutting a piece of construction paper in the shape of a parallelogram. Two opposite sides of the parallelogram have lengths [tex]\((5n - 6)\)[/tex] cm and [tex]\((3n - 2)\)[/tex] cm. A third side measures [tex]\((2n + 3)\)[/tex] cm.

What are the lengths of two adjacent sides of the parallelogram?

A. 2 cm and 2 cm
B. 4 cm and 7 cm
C. 7 cm and 9 cm
D. 13 cm and 19 cm



Answer :

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]