To determine the lengths of two adjacent sides of the parallelogram, we need to find the value of [tex]\( n \)[/tex] that makes the given side lengths equal.
The information provided gives us two expressions for the lengths of the sides of the parallelogram:
- One side has a length of [tex]\( 5n - 6 \)[/tex] cm.
- The opposite side has a length of [tex]\( 3n - 2 \)[/tex] cm.
- A third side has a length of [tex]\( 2n + 3 \)[/tex] cm.
In a parallelogram, opposite sides are equal. Thus, we equate the expressions for the two opposite sides and solve for [tex]\( n \)[/tex]:
[tex]\[ 5n - 6 = 3n - 2 \][/tex]
Now, solve this equation step-by-step:
1. Combine like terms by subtracting [tex]\( 3n \)[/tex] from both sides of the equation:
[tex]\[ 5n - 3n - 6 = 3n - 3n - 2 \][/tex]
[tex]\[ 2n - 6 = -2 \][/tex]
2. Add 6 to both sides of the equation:
[tex]\[ 2n - 6 + 6 = -2 + 6 \][/tex]
[tex]\[ 2n = 4 \][/tex]
3. Divide both sides by 2 to isolate [tex]\( n \)[/tex]:
[tex]\[ \frac{2n}{2} = \frac{4}{2} \][/tex]
[tex]\[ n = 2 \][/tex]
Now that we have the value of [tex]\( n \)[/tex], we can substitute it back into the expressions for the side lengths to determine their actual lengths:
1. For the first side [tex]\( 5n - 6 \)[/tex]:
[tex]\[ 5(2) - 6 = 10 - 6 = 4 \][/tex]
2. For the opposite side [tex]\( 3n - 2 \)[/tex]:
[tex]\[ 3(2) - 2 = 6 - 2 = 4 \][/tex]
3. For the third side [tex]\( 2n + 3 \)[/tex]:
[tex]\[ 2(2) + 3 = 4 + 3 = 7 \][/tex]
So we find that the lengths of two adjacent sides of the parallelogram are 4 cm and 7 cm. Thus, the correct answer is:
4 cm and 7 cm
