Juanita Is Cutting A Piece Of Construction Paper In The Shape Of A Parallelogram. Two Opposite Sides Have Lengths \[$(5n-6)\$\] Cm And \[$(3n-2)\$\] Cm. A Third Side Measures \[$(2n+3)\$\] Cm.What Are The Lengths Of Two Adjacent

Feb 28, 2025 by ADMIN 229 views

**Juanita's Parallelogram Puzzle: Unraveling the Mystery of Opposite Sides**

Introduction

Juanita is busy cutting a piece of construction paper in the shape of a parallelogram. A parallelogram is a quadrilateral with opposite sides that are parallel to each other. In this problem, we are given the lengths of two opposite sides and a third side. Our goal is to find the lengths of the two adjacent sides. To tackle this problem, we need to understand the properties of parallelograms and use algebraic techniques to solve for the unknown side lengths.

Properties of Parallelograms

A parallelogram has several key properties that we can use to our advantage. These properties include:

Opposite sides are parallel: This means that if we draw a line connecting the opposite vertices of the parallelogram, it will be a straight line.
Opposite sides are equal in length: This means that if we measure the length of one side, the length of the opposite side will be the same.
Consecutive angles are supplementary: This means that if we draw a diagonal line connecting two opposite vertices, the two angles formed by this diagonal will add up to 180 degrees.

Given Information

We are given the following information about the parallelogram:

Two opposite sides have lengths ${$ (5n-6)$}$ cm and ${$ (3n-2)$}$ cm.
A third side measures ${$ (2n+3)$}$ cm.

Finding the Lengths of Adjacent Sides

To find the lengths of the two adjacent sides, we can use the properties of parallelograms. Since opposite sides are equal in length, we know that the length of the fourth side is also ${$ (5n-6)$}$ cm.

Now, let's consider the third side, which measures ${$ (2n+3)$}$ cm. Since this side is adjacent to the two opposite sides, we can use the fact that consecutive angles are supplementary to find the length of the fourth side.

Using Algebraic Techniques

To solve for the length of the fourth side, we can use algebraic techniques. Let's call the length of the fourth side ${$ x$}$ cm. Since opposite sides are equal in length, we know that ${$ x = 5n-6$}$.

Using the fact that consecutive angles are supplementary, we can write an equation:

${$ (2n+3) + x = 180$}$

Simplifying this equation, we get:

${ $2n + 3 + 5n - 6 = 180\$}$

Combine like terms:

${ $7n - 3 = 180\$}$

Add 3 to both sides:

${ $7n = 183\$}$

Divide both sides by 7:

${$ n = 26.14$}$

Now that we have found the value of ${$ n$}$, we can substitute it into the expressions for the side lengths:

The length of the first side is ${$ (5n-6) = (5(26.14)-6) = 130.7$}$ cm.
The length of the second side is ${$ (3n-2) = (3(26.14)-2) = 77.42$}$ cm.
The length of the third side is ${$ (2n+3) = (2(26.14)+3) = 55.28$}$ cm.
The length of the fourth side is ${$ (5n-6) = (5(26.14)-6) = 130.7$}$ cm.

Conclusion

In this problem, we used the properties of parallelograms and algebraic techniques to find the lengths of the two adjacent sides. We found that the length of the first side is ${$ (5n-6) = 130.7$}$ cm, the length of the second side is ${$ (3n-2) = 77.42$}$ cm, the length of the third side is ${$ (2n+3) = 55.28$}$ cm, and the length of the fourth side is ${$ (5n-6) = 130.7$}$ cm.

Final Answer

The final answer is:

The length of the first side is ${$ (5n-6) = 130.7$}$ cm.
The length of the second side is ${$ (3n-2) = 77.42$}$ cm.
The length of the third side is ${$ (2n+3) = 55.28$}$ cm.
The length of the fourth side is ${$ (5n-6) = 130.7$}$ cm.
Juanita's Parallelogram Puzzle: Unraveling the Mystery of Opposite Sides ====================================================================================

Q&A: Frequently Asked Questions

Q: What is a parallelogram?

A: A parallelogram is a quadrilateral with opposite sides that are parallel to each other.

Q: What are the properties of parallelograms?

A: The properties of parallelograms include:

Opposite sides are parallel: This means that if we draw a line connecting the opposite vertices of the parallelogram, it will be a straight line.
Opposite sides are equal in length: This means that if we measure the length of one side, the length of the opposite side will be the same.
Consecutive angles are supplementary: This means that if we draw a diagonal line connecting two opposite vertices, the two angles formed by this diagonal will add up to 180 degrees.

Q: How do we find the lengths of adjacent sides in a parallelogram?

A: To find the lengths of adjacent sides in a parallelogram, we can use the properties of parallelograms. Since opposite sides are equal in length, we know that the length of the fourth side is also equal to the length of the first side.

Q: What is the formula for finding the length of the fourth side in a parallelogram?

A: The formula for finding the length of the fourth side in a parallelogram is:

${$ x = 5n-6$}$

where ${$ x$}$ is the length of the fourth side and ${$ n$}$ is a variable.

Q: How do we use algebraic techniques to solve for the length of the fourth side?

A: To solve for the length of the fourth side, we can use algebraic techniques. Let's call the length of the fourth side ${$ x$}$ cm. Since opposite sides are equal in length, we know that ${$ x = 5n-6$}$.