Parallelogram ABCD: Solving For Side Length With Area & Angle

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Hey math enthusiasts! Let's dive into a geometry problem that's both engaging and practical. We're given a parallelogram named ABCD, and we're going to use some clever tricks to figure out the length of one of its sides. This isn't just about numbers; it's about understanding how shapes work and how we can use formulas to uncover hidden information. So, grab your pens and paper, and let's get started. We'll break down this problem step by step, making sure everyone can follow along. By the end, you'll be able to confidently tackle similar problems. Let's begin by understanding what a parallelogram is, then we will dive deep into the problem. First, we have to know what we have.

Understanding the Basics: What's a Parallelogram?

Alright, before we jump into the numbers, let's make sure we're all on the same page about parallelograms. A parallelogram is a four-sided shape (a quadrilateral) where the opposite sides are parallel and equal in length. Think of it like a slanted rectangle; it has two pairs of parallel sides, but its angles aren't necessarily right angles. This is super important because it affects how we calculate its area and the relationships between its sides and angles. Key things to remember: Opposite sides are parallel and equal, opposite angles are equal, and adjacent angles are supplementary (they add up to 180 degrees).

This knowledge is the foundation. Remember, the area of a parallelogram is base times height (Area = base * height). The height is the perpendicular distance between the base and its opposite side. The fact that we're dealing with an angle (∠ABC = 30°) means we'll likely need some trigonometry, too. Don't worry, it's not as scary as it sounds. It's just a way to relate the angles and sides in a right-angled triangle. Once we have those pieces, we can start to formulate our solution and come closer to the answer.

To successfully work through this problem, we need to be confident in our math skills, especially in how parallelograms work. This knowledge is the key to unlock our problem. So, let's review some key facts: Opposite sides are equal in length. The opposite angles are equal. Adjacent angles add up to 180 degrees. Understanding these fundamentals, together with the area formula, will help us find our missing side. Remember, the height is the distance from one side to the other. So, if you understand these concepts, then you are on the right track.

Core Properties of Parallelograms

Here's a quick recap of the core properties that make parallelograms unique and useful in geometry:

  1. Parallel Sides: By definition, a parallelogram has two pairs of parallel sides. This is the most fundamental property, and it dictates many other characteristics.
  2. Equal Opposite Sides: The sides opposite each other in a parallelogram are not just parallel; they are also equal in length. This means AB = CD and BC = DA in our problem.
  3. Equal Opposite Angles: The angles opposite each other in a parallelogram are congruent. So, ∠ABC = ∠CDA and ∠BAD = ∠BCD.
  4. Supplementary Adjacent Angles: Adjacent angles (angles next to each other) are supplementary, meaning they add up to 180 degrees. For example, ∠ABC + ∠BAD = 180°.
  5. Diagonals Bisect Each Other: The diagonals of a parallelogram (lines connecting opposite vertices) bisect each other. This means they cut each other in half at their intersection point.
  6. Area Formula: The area of a parallelogram is calculated using the formula: Area = base × height, where the height is the perpendicular distance between the base and its opposite side.

These properties are not just abstract concepts; they are the tools we'll use to solve our problem. They provide the relationships and constraints that allow us to find the missing side length. As we move forward, remember these properties. They will guide us through the process.

Breaking Down the Problem: The Given Information

Now that we know what a parallelogram is, let's look at what we're given. We're told that the area of parallelogram ABCD is 420 cm². The base AB is 28 cm, and angle ABC is 30°. Our mission? To find the length of the other side of the parallelogram. This is where we need to combine our knowledge of parallelograms with some basic trigonometry to find the other side.

Here's a summary of what we have: Area = 420 cm², Base (AB) = 28 cm, ∠ABC = 30°. We need to find the length of BC (or AD, since they are equal in a parallelogram). The question does not seem hard. It's like a treasure hunt! Our clues lead us towards finding the other side. We know the area, base, and one angle. We can determine a height, then we will be able to find the other side of the parallelogram. This approach ensures we don't miss anything and helps us build a complete solution.

Step-by-Step Approach to Solving the Problem

Here is how we will solve this step by step:

  1. Visualize: Always start by drawing a diagram of the parallelogram. Label the sides and the angle. This visual aid helps in understanding the relationships between the sides and the angles.
  2. Find the Height: Use the area and base to find the height. The area of a parallelogram is calculated as Area = base × height. We know the area and base, so we can solve for the height.
  3. Use Trigonometry: Since we know the angle ABC (30°) and the height, we can use trigonometry (specifically the sine function) to find the length of BC. Remember, the sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.
  4. Calculate the Side Length: After finding the height, we'll use the sine function to calculate the length of side BC. This will be our final answer.

This method simplifies a complex problem by breaking it down into manageable pieces. It’s also an approach that promotes better understanding and retention of key concepts. So, stay with me as we go step by step!

Finding the Height of the Parallelogram

Alright, let's use the formula for the area of a parallelogram: Area = base × height. We know the area (420 cm²) and the base (28 cm). So, we can rearrange the formula to solve for the height: height = Area / base. Let's plug in the values: height = 420 cm² / 28 cm = 15 cm. The height of the parallelogram is 15 cm. This is the perpendicular distance between the base AB and the opposite side CD. The height is a crucial element of our solution.

Why is the height so important? It forms a right-angled triangle with the side BC and a part of the base. This right triangle gives us a place to use trigonometry, specifically the sine function. Without the height, we couldn’t easily relate the angle ABC (30°) to the side lengths. The height is the key that unlocks the next step, allowing us to find the length of the side BC. Now we can start using our trigonometry knowledge. We have the angle, and we have the height. Let's take a closer look. Having the height enables us to connect the given angle to the sides, forming a right-angled triangle. This is the perfect setup for using trigonometric functions, and in this case, the sine function. The height enables us to relate the angle, the height, and the side BC in a clean, straightforward manner.

Detailed Steps for Calculating Height

  1. Formula: Start with the area formula: Area = base × height.
  2. Substitute Values: We know Area = 420 cm² and base = 28 cm. So, 420 = 28 × height.
  3. Solve for Height: Divide both sides by 28: height = 420 / 28 = 15 cm.
  4. Result: The height of the parallelogram is 15 cm.

Knowing the height allows us to use the angle ABC to find the side. Let's use the sine function.

Using Trigonometry to Find the Side Length

Now, we have the height of the parallelogram and the angle ABC. We can use trigonometry to find the length of the side BC. Specifically, we will use the sine function. In a right-angled triangle (formed by the height, a part of the base, and the side BC), the sine of an angle is the ratio of the opposite side (the height) to the hypotenuse (side BC). So, sin(30°) = height / BC.

We know that sin(30°) = 0.5 (you can find this in a trig table or calculator). We also know the height is 15 cm. So, we can set up the equation: 0.5 = 15 cm / BC. To solve for BC, rearrange the equation: BC = 15 cm / 0.5 = 30 cm. Therefore, the length of the other side (BC) of the parallelogram is 30 cm. This final step is the culmination of our work. By using trigonometry and the area, we have successfully found the length of the unknown side.

Detailed Steps Using Trigonometry

  1. Draw a Right Triangle: Imagine or draw a right-angled triangle within the parallelogram, with the height as one side, a part of the base as the other side, and BC as the hypotenuse.
  2. Identify the Sine Ratio: The sine of the angle (30°) is the ratio of the height (opposite side) to the hypotenuse (BC).
  3. Substitute Values: We know sin(30°) = 0.5, and the height = 15 cm. So, 0.5 = 15 cm / BC.
  4. Solve for BC: Rearrange the equation: BC = 15 cm / 0.5 = 30 cm.
  5. Final Answer: BC = 30 cm.

Conclusion: The Other Side Revealed!

And there you have it! We've successfully found the length of the other side of the parallelogram ABCD. By understanding the properties of parallelograms, calculating the height, and applying trigonometry, we discovered that the length of side BC is 30 cm.

This problem beautifully illustrates how different areas of mathematics come together to solve a single problem. It's not just about formulas; it's about understanding the relationships between the parts of a shape and using that knowledge to find the answers. We took the first step to solve the problem by understanding the properties of a parallelogram. Then, we calculated the height using the area formula. After that, we applied trigonometry to find the length of the unknown side. This problem provides a good opportunity to review. Remember, geometry can be fun and understandable.

So, next time you encounter a geometry problem, don't be intimidated. Break it down into manageable steps, use the formulas, and remember to visualize the problem. You've got this!

Summary of Key Findings

  • Area of Parallelogram: 420 cm².
  • Base (AB): 28 cm.
  • Angle ABC: 30°.
  • Height: 15 cm (calculated).
  • Side BC (other side): 30 cm (calculated using trigonometry).

This detailed breakdown should give you the clarity you need to understand the problem and the solution. Keep practicing, and you'll become a geometry pro in no time! I hope you enjoyed the journey of solving the problem. Keep learning, and you will improve your skills.