Rectangular Trapezoid ABCD: Calculate AD Length

Hey guys! Today, we're diving into a geometry problem that involves a rectangular trapezoid. Let's break it down step by step so we can all understand it clearly. We're going to calculate the length of the major base AD in a rectangular trapezoid ABCD where the perpendicular side AB is twice the length of the shorter base BC, the oblique side CD measures 12 cm, and angle D is 60°. Sounds like a fun challenge, right? Let’s jump in!

Understanding the Problem: Setting Up the Basics

Okay, so first things first, let's visualize what we're dealing with. A rectangular trapezoid, in our case ABCD, has a couple of key features. It has two parallel sides (bases) and two non-parallel sides. The 'rectangular' part means that at least one of the non-parallel sides is perpendicular to the bases, forming a right angle. In our scenario, side AB is perpendicular to both bases BC and AD, making angles A and B right angles. The problem tells us that side AB (the perpendicular side) is twice as long as the shorter base BC. We also know the oblique side CD is 12 cm and the angle D is 60 degrees. Our mission? To find the length of AD, the longer base.

To really nail this, sketching out the trapezoid can be super helpful. Draw a vertical line for AB, then a shorter horizontal line for BC extending from the top of AB. AD, the longer base, will extend from the bottom of AB, and CD will connect the end of BC to the end of AD, forming that 60-degree angle at D. This visual representation allows us to see the relationships between the sides and angles more clearly. We can start thinking about how we can use this information to find the length of AD. One way to tackle this is by breaking down the trapezoid into simpler shapes, like rectangles and triangles. This will allow us to use basic geometric principles and trigonometric ratios to find our missing length. So, grab a pen and paper, and let’s get sketching!

Breaking Down the Trapezoid: Geometry to the Rescue

Now that we have a visual, let’s break this trapezoid down into shapes we know and love – a rectangle and a right-angled triangle. Imagine drawing a vertical line from point C down to AD, and let’s call the point where it meets AD point E. What we’ve effectively done is split the trapezoid ABCD into rectangle ABCE and right-angled triangle CDE. This is a classic problem-solving strategy in geometry, guys! When faced with a complex shape, decompose it into simpler shapes whose properties we understand.

In our new setup, ABCE is a rectangle, which means AB = CE and BC = AE. This is super helpful because we know AB is twice the length of BC. Let's say BC has a length of 'x'. That means AB is '2x'. Since AB = CE, we know CE is also '2x'. Now, let's shift our focus to the right-angled triangle CDE. We know the hypotenuse CD is 12 cm, and angle D is 60 degrees. We also know the side CE, which is opposite to angle D, has a length of '2x'. To solve for x, we can use our trigonometric friends, specifically the sine function. Remember, sin(angle) = opposite/hypotenuse. So, sin(60°) = CE/CD, which translates to sin(60°) = 2x/12. Once we find the value of x, we’re one step closer to finding AD. Hang in there, we’re making progress!

Trigonometry to the Rescue: Calculating the Unknown

Alright, let’s put our trigonometry hats on and dive into solving for 'x'. We’ve established that sin(60°) = 2x/12. Now, we need to remember our special trigonometric values. The sine of 60 degrees, sin(60°), is equal to √3/2 (that's the square root of 3 divided by 2). So, our equation becomes √3/2 = 2x/12. To solve for x, we can cross-multiply and simplify. Multiplying both sides by 12 gives us 12 * (√3/2) = 2x, which simplifies to 6√3 = 2x. Now, divide both sides by 2 to isolate x, and we get x = 3√3. Awesome! We’ve found the value of x, which is the length of BC. But remember, we’re trying to find the length of AD, the longer base.

Now that we know x, we can figure out the lengths of other segments in our shapes. We know BC = x = 3√3, and since ABCE is a rectangle, AE = BC = 3√3 as well. We also need to find the length of ED to calculate AD. Back to our right-angled triangle CDE! We know CD is 12 cm and CE is 2x, which is 2 * 3√3 = 6√3 cm. To find ED, we can use either the Pythagorean theorem or another trigonometric ratio. Let’s use the cosine function this time. Cosine is adjacent over hypotenuse, so cos(60°) = ED/CD. The cosine of 60 degrees, cos(60°), is 1/2. So, 1/2 = ED/12. Multiply both sides by 12, and we get ED = 6 cm. We're almost there! Now we have all the pieces we need to calculate AD.

Putting It All Together: Finding the Length of AD

Okay, guys, it’s time to piece everything together and find the length of AD. We’ve broken down the trapezoid, used trigonometry, and calculated the necessary side lengths. We know that AD is made up of two segments: AE and ED. We’ve already found that AE = BC = 3√3 cm, and ED = 6 cm. So, to find AD, we simply add AE and ED together: AD = AE + ED = 3√3 + 6 cm. And there we have it!

So, the length of the major base AD is 3√3 + 6 cm. That’s the final answer! We’ve successfully navigated through this geometry problem by breaking it down into smaller, more manageable parts. We visualized the problem, decomposed the trapezoid into a rectangle and a right-angled triangle, used trigonometric ratios to find unknown lengths, and finally, combined these lengths to find the length of AD. Remember, guys, the key to solving complex geometry problems is to break them down, use what you know, and take it one step at a time. Great job, everyone!

Conclusion: Geometry Mastery Achieved

Wow, what a journey through the world of trapezoids and trigonometry! We successfully found the length of the longer base AD in our rectangular trapezoid ABCD. Remember, the approach we used here – breaking down complex shapes into simpler ones, applying trigonometric principles, and using basic geometric relationships – is a powerful strategy for tackling all sorts of geometry problems. By understanding these fundamental concepts and practicing regularly, you'll be able to confidently solve even the trickiest geometric challenges. Keep up the great work, and keep exploring the fascinating world of mathematics! You guys rock!