Area Of Quadrilateral ABCD: A Step-by-Step Solution

Hey guys! Let's break down how to find the area of quadrilateral ABCD. This is a classic geometry problem, and we'll go through it step by step to make sure everyone understands. We're given that side AD is 20 cm, side DC is also 20 cm, and the angle between them is a right angle (90 degrees). Plus, we have some multiple-choice answers to guide us. Let's dive in!

Understanding the Problem

Before we start crunching numbers, let's visualize what we're dealing with. We have a quadrilateral ABCD, where two sides, AD and DC, meet at a 90-degree angle. This is a big clue because it suggests that part of our quadrilateral might be a familiar shape – a right-angled triangle or even a square. Recognizing these shapes will help us simplify the problem and find the area more easily.

Why is visualization important? Well, geometry isn't just about formulas; it's about seeing the shapes and understanding their relationships. When you sketch the problem or imagine it clearly, you often spot shortcuts or simpler ways to solve it. So, always start by visualizing!

Now, let's put that visualization into action. Imagine drawing the quadrilateral. You've got AD going in one direction, then DC making a sharp right turn. What we need to figure out is how the other two sides, AB and BC, complete the shape. Without more information about these sides, we can't determine a unique quadrilateral. But hold on! The problem probably intends for us to assume something simpler. Let’s explore that next.

Making Assumptions and Simplifying

Okay, so the problem only gives us information about two sides and one angle. This suggests we should make a simplifying assumption. The most logical one here is to assume that the quadrilateral ABCD is actually a square. Why? Because if AD = 20 cm, DC = 20 cm, and the angle ADC is 90 degrees, then completing the square is the simplest way to close the figure. If ABCD is a square, then AB = BC = 20 cm, and all angles are 90 degrees.

Why is this a reasonable assumption? In math problems, especially in multiple-choice scenarios, the simplest interpretation is often the correct one unless there's information suggesting otherwise. If the problem intended a more complex quadrilateral, it would likely provide more details about sides AB and BC, or angles ABC and BCD. Since we don't have that, let's roll with the square assumption.

However, it's super important to note that without additional information, the shape could be something else. For instance, we could have a trapezoid or an irregular quadrilateral. But in those cases, we'd need more measurements or angle information to calculate the area. Given the limited information and the multiple-choice format, assuming a square is the most straightforward and likely correct approach.

Calculating the Area

Alright, assuming we're dealing with a square, calculating the area becomes super easy. The area of a square is simply the side length squared. In our case, the side length is 20 cm. So, here's the calculation:

Area = side × side = 20 cm × 20 cm = 400 cm²

Boom! That's it. If our assumption is correct (and given the problem's simplicity, it probably is), the area of quadrilateral ABCD is 400 square centimeters.

Now, let's quickly check our answer against the multiple-choice options:

A) 200 cm² B) 400 cm² C) 300 cm² D) 250 cm²

Our calculated answer matches option B. So, we're feeling pretty confident about this.

Addressing Potential Complications

Okay, but what if the quadrilateral isn't a perfect square? What if it's something a bit more… wonky? Well, in that case, we'd need more information to solve the problem. For instance, if we knew the lengths of sides AB and BC, and the angles at vertices B and C, we could potentially divide the quadrilateral into simpler shapes (like triangles) and calculate their areas separately. Then, we'd add those areas together to find the total area of the quadrilateral.

But here's the thing: without that extra information, there's no way to determine a unique area for the quadrilateral. It could be anything! That's why, given the problem's constraints, assuming a square is the most reasonable and practical approach.

To illustrate, imagine tilting side AB. The area changes. Now, imagine tilting side BC. Again, the area shifts. Without knowing the specific angles and side lengths, calculating the area is just not possible.

Final Answer and Justification

So, after walking through the problem, making a reasonable assumption, and doing the math, our final answer is:

B) 400 cm²

Why are we so confident in this answer? Because:

1. The problem provides limited information, suggesting a simple solution is intended.
2. Assuming a square is the most straightforward interpretation of the given data.
3. Our calculation based on the square assumption matches one of the multiple-choice options.
4. Without additional information, determining a unique area for a non-square quadrilateral is impossible.

Therefore, based on the available information and the nature of the question, option B is the most logical and defensible answer. Remember always think simple and visualize!