Parallelogram Dimensions Solving For Adjacent Sides

Introduction: Unveiling the Dimensions of a Parallelogram

In this intriguing mathematical puzzle, we delve into the world of geometry, specifically parallelograms. Our focus is on Juanita, who is diligently cutting a piece of construction paper into the shape of a parallelogram. The problem presents us with algebraic expressions representing the lengths of the parallelogram's sides. Two opposite sides are given as $(5n-6)$ cm and $(3n-2)$ cm, while a third side measures $(2n+3)$ cm. Our primary objective is to determine the lengths of two adjacent sides of this parallelogram. This seemingly simple problem requires us to understand the properties of parallelograms, apply algebraic principles to solve for the unknown variable 'n', and finally, calculate the side lengths. Understanding the fundamental properties of parallelograms is crucial in solving this problem. A parallelogram, by definition, is a quadrilateral with two pairs of parallel sides. A key property that we will utilize is that opposite sides of a parallelogram are equal in length. This property allows us to set up an equation based on the given side lengths. Additionally, the concept of adjacent sides is important. Adjacent sides are sides that share a common vertex, meaning they are next to each other. In our problem, we need to find the lengths of two such adjacent sides. The algebraic expressions provided, $(5n-6)$ cm, $(3n-2)$ cm, and $(2n+3)$ cm, introduce the variable 'n'. To find the numerical lengths of the sides, we must first determine the value of 'n'. This involves setting up an equation based on the properties of parallelograms and solving for 'n' using algebraic techniques. The equation will stem from the fact that opposite sides of a parallelogram are equal. Once we have the value of 'n', we can substitute it back into the expressions for the side lengths to find their numerical values. This step involves basic arithmetic operations. Finally, we will identify two adjacent sides and state their lengths as the solution to the problem. This requires careful consideration of which sides are next to each other in the parallelogram. Let's embark on this geometrical journey and unravel the dimensions of Juanita's parallelogram.

Solving for 'n': Leveraging Parallelogram Properties

To solve this geometric puzzle, we must first decipher the value of 'n'. The key to unlocking this lies in understanding a fundamental property of parallelograms: opposite sides are equal in length. This property allows us to establish a crucial equation. We are given that two opposite sides have lengths $(5n-6)$ cm and $(3n-2)$ cm. Since opposite sides are equal, we can confidently equate these two expressions: $(5n - 6) = (3n - 2)$. Now, we have a simple algebraic equation that we can solve for 'n'. To isolate 'n', we will first subtract $3n$ from both sides of the equation. This yields: $5n - 3n - 6 = 3n - 3n - 2$, which simplifies to $2n - 6 = -2$. Next, we need to isolate the term with 'n' further. To do this, we add 6 to both sides of the equation: $2n - 6 + 6 = -2 + 6$. This simplifies to $2n = 4$. Finally, to solve for 'n', we divide both sides of the equation by 2: $\frac{2n}{2} = \frac{4}{2}$. This gives us the solution: $n = 2$. Now that we have determined the value of 'n', we can proceed to calculate the lengths of the parallelogram's sides. This is a crucial step, as it bridges the gap between the algebraic expressions and the actual numerical dimensions of the shape. The value of 'n' acts as a key that unlocks the specific lengths of the sides, allowing us to move closer to the final solution of the problem. We will substitute this value back into the expressions provided in the problem statement to find the individual side lengths. This process will not only provide us with numerical answers but also reinforce the connection between algebraic representation and geometric reality. The next step involves substituting $n = 2$ into the expressions for the side lengths, which will reveal the concrete dimensions of Juanita's parallelogram. This will bring us closer to identifying the lengths of the two adjacent sides, which is the ultimate goal of the problem.

Calculating Side Lengths: Substituting 'n' to Find Dimensions

With the value of 'n' successfully determined to be 2, we can now embark on the crucial step of calculating the lengths of the parallelogram's sides. This involves substituting $n = 2$ into the algebraic expressions provided in the problem. Let's begin by considering the sides with lengths expressed as $(5n - 6)$ cm and $(3n - 2)$ cm. These sides, as we established earlier, are opposite each other and therefore should have equal lengths. Substituting $n = 2$ into the first expression, we get: $(5 * 2 - 6) = (10 - 6) = 4$ cm. Similarly, substituting $n = 2$ into the second expression, we get: $(3 * 2 - 2) = (6 - 2) = 4$ cm. As expected, both expressions yield the same length, 4 cm, confirming our calculations and reinforcing the property of parallelograms that opposite sides are equal. Now, let's consider the third side, which has a length of $(2n + 3)$ cm. Substituting $n = 2$ into this expression, we get: $(2 * 2 + 3) = (4 + 3) = 7$ cm. This gives us the length of the side adjacent to the sides we calculated earlier. Now that we have calculated the lengths of all three sides based on the provided expressions, we can confidently state the dimensions of Juanita's parallelogram. We have two sides that measure 4 cm each and another side that measures 7 cm. Remember that in a parallelogram, there are two pairs of equal sides. Therefore, we now know that the other side adjacent to the 4 cm side will also measure 7 cm. This completes our understanding of the parallelogram's dimensions. The next and final step is to identify the lengths of two adjacent sides. We have the lengths of all the sides, so this is a straightforward task. Adjacent sides are those that share a vertex. In our case, we have sides of length 4 cm and 7 cm. These sides are adjacent to each other, forming a corner of the parallelogram. Therefore, the solution to our problem is that the two adjacent sides have lengths 4 cm and 7 cm. This completes the puzzle, providing us with the specific dimensions of the parallelogram that Juanita is cutting.

Identifying Adjacent Sides: The Final Piece of the Puzzle

Having calculated the lengths of all the sides of Juanita's parallelogram, we now stand at the final step: identifying the lengths of two adjacent sides. Recall that adjacent sides are those that share a common vertex, essentially forming a corner of the parallelogram. We have determined that two sides of the parallelogram have a length of 4 cm each, and the other two sides have a length of 7 cm each. Therefore, any side measuring 4 cm will be adjacent to a side measuring 7 cm. Visualize the parallelogram: you can imagine the sides of 4 cm and 7 cm meeting at a corner. These are our adjacent sides. This understanding of adjacency is key to providing the final answer to the problem. We have successfully navigated through the algebraic manipulations and geometric principles to arrive at this point. Now, we can definitively state the solution. The lengths of two adjacent sides of Juanita's parallelogram are 4 cm and 7 cm. This concludes our exploration of this geometric puzzle. We started with algebraic expressions representing the side lengths, utilized the properties of parallelograms to solve for the unknown variable, calculated the numerical side lengths, and finally, identified the lengths of two adjacent sides. This process demonstrates the power of combining algebraic techniques with geometric understanding to solve real-world problems. The problem not only reinforces the properties of parallelograms but also highlights the importance of algebraic manipulation in solving geometric problems. By breaking down the problem into smaller, manageable steps, we were able to systematically arrive at the solution. The final answer, 4 cm and 7 cm, represents the culmination of our efforts and a clear understanding of the relationships within a parallelogram. This journey through Juanita's parallelogram serves as a valuable exercise in mathematical problem-solving, showcasing the interconnectedness of algebra and geometry. We have not only found the answer but also deepened our understanding of geometric shapes and their properties.

Conclusion: The Solution to Juanita's Parallelogram

In conclusion, after a thorough exploration of Juanita's parallelogram, we have successfully determined the lengths of two adjacent sides. By leveraging the fundamental properties of parallelograms, particularly the equality of opposite sides, we were able to set up and solve an algebraic equation to find the value of 'n'. This crucial step allowed us to bridge the gap between the algebraic representation of the side lengths and their numerical values. Substituting the value of 'n' back into the expressions, we calculated the side lengths to be 4 cm and 7 cm. This revealed the specific dimensions of Juanita's parallelogram, bringing the abstract geometric concept to a concrete realization. Finally, understanding the concept of adjacency, we confidently identified the two adjacent sides as having lengths of 4 cm and 7 cm. This concludes our journey through this geometric puzzle, demonstrating the power of mathematical reasoning and problem-solving skills. The solution not only provides the answer to the specific question but also reinforces our understanding of parallelograms and their properties. The process involved a combination of algebraic manipulation and geometric insight, highlighting the interconnectedness of these two branches of mathematics. We started with an abstract problem, carefully analyzed the given information, applied relevant theorems and principles, and systematically arrived at a concrete solution. This exemplifies the essence of mathematical problem-solving. The lengths of the adjacent sides, 4 cm and 7 cm, represent the final piece of the puzzle. Juanita's parallelogram now stands fully defined, a testament to our analytical and problem-solving abilities. This exercise not only enhances our mathematical skills but also cultivates our ability to approach complex problems with a structured and logical approach. The successful completion of this problem provides a sense of accomplishment and reinforces the value of mathematical knowledge in understanding and solving real-world scenarios. The journey through Juanita's parallelogram serves as a valuable lesson in geometric problem-solving, leaving us with a deeper appreciation for the beauty and power of mathematics.

Therefore, the lengths of the two adjacent sides of Juanita's parallelogram are 4 cm and 7 cm.