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

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

A. [tex]\(2 \, \text{cm}\)[/tex] and [tex]\(2 \, \text{cm}\)[/tex]
B. [tex]\(4 \, \text{cm}\)[/tex] and [tex]\(7 \, \text{cm}\)[/tex]
C. [tex]\(7 \, \text{cm}\)[/tex] and [tex]\(9 \, \text{cm}\)[/tex]
D. [tex]\(13 \, \text{cm}\)[/tex] and [tex]\(19 \, \text{cm}\)[/tex]



Answer :

GinnyAnswer
To determine the lengths of two adjacent sides of the parallelogram, we need to follow these steps:

1. Understand the properties of a parallelogram:
- Opposite sides of a parallelogram are equal in length.
- Hence, we can set the expressions for opposite sides equal to each other and solve for [tex]\( n \)[/tex].

2. Given expressions for the lengths of two opposite sides:
- One side length: [tex]\( 5n - 6 \)[/tex] cm
- Opposite side length: [tex]\( 3n - 2 \)[/tex] cm

3. Set the expressions for these sides equal to solve for [tex]\( n \)[/tex]:
[tex]\[ 5n - 6 = 3n - 2 \][/tex]

4. Solve the equation for [tex]\( n \)[/tex]:
- Subtract [tex]\( 3n \)[/tex] from both sides:
[tex]\[ 2n - 6 = -2 \][/tex]
- Add 6 to both sides:
[tex]\[ 2n = 4 \][/tex]
- Divide both sides by 2:
[tex]\[ n = 2 \][/tex]

5. Use the value of [tex]\( n \)[/tex] to calculate the lengths of the sides:
- Substitute [tex]\( n = 2 \)[/tex] into each expression:
[tex]\[ \text{Side 1: } 5n - 6 = 5(2) - 6 = 10 - 6 = 4 \, \text{cm} \][/tex]
[tex]\[ \text{Side 3: } 2n + 3 = 2(2) + 3 = 4 + 3 = 7 \, \text{cm} \][/tex]

6. Verify the length of the other pair of opposite sides:
- We already know that [tex]\( 3n - 2 \)[/tex] should be equal to [tex]\( 4 \, \text{cm} \)[/tex]:
[tex]\[ \text{Side 2: } 3n - 2 = 3(2) - 2 = 6 - 2 = 4 \, \text{cm} \][/tex]

So the lengths of the two adjacent sides of the parallelogram are [tex]\(4\)[/tex] cm and [tex]\(7\)[/tex] cm.

Hence, the correct pair of adjacent sides is:
[tex]\[ \boxed{4 \text{ cm and } 7 \text{ cm}} \][/tex]

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