Prism Diagonal & Radius: Prove Lateral Edge Is 5cm

Hey guys! Let's dive into a cool geometry problem today. We've got a right regular quadrangular prism, some diagonals, and a circumscribed circle. Our mission? To show that the lateral edge of this prism is a neat 5 cm. Ready to roll up your sleeves and get into some math? Let’s break it down together!

Understanding the Right Regular Quadrangular Prism

First off, what exactly is a right regular quadrangular prism? It sounds like a mouthful, but let's dissect it. Think of a box, but with some specific rules. The term "quadrangular" tells us that the bases are quadrilaterals—four-sided shapes. Now, "regular" means that these quadrilaterals are actually squares. So, we're dealing with a prism where the top and bottom are perfect squares. And what about "right"? This means the lateral faces (the sides) are rectangles, and they stand upright, forming right angles with the bases. Imagine a perfectly upright box with square ends, and you’ve got it!

So, in our case, we have prism ABCDA'B'C'D'. The bases, ABCD and A'B'C'D', are squares. The sides, like ABB'A', are rectangles. And we know that all those angles where the sides meet the bases are 90 degrees. This is super important because right angles open the door to using the Pythagorean Theorem, a key tool in our geometric toolbox. Knowing this helps us visualize the problem and strategize our approach. Now, before we jump into the calculations, let's make sure we understand all the given information. We know the diagonal of the prism is 13 cm, and the radius of the circle circumscribed around the base is 6 cm. This radius is essentially the distance from the center of the square base to one of its corners. We need to connect these pieces of information to find the length of the lateral edge, which is the height of the prism. It’s like fitting puzzle pieces together to reveal the bigger picture!

Decoding the Given Information

Now, let's really dig into the information we've been given. We know the diagonal of the prism is 13 cm. But what diagonal are we talking about here? In a prism, there are a few types of diagonals, but we're most likely talking about the space diagonal, which runs from one corner of the bottom base to the opposite corner of the top base (like AC'). This is crucial because it forms a right triangle with the diagonal of the base and the lateral edge of the prism. Think of it as the longest straight line you can draw inside the prism, connecting two far-off corners. This space diagonal is our hypotenuse in a 3D right triangle, and that's super helpful!

Next, we have the radius of the circle circumscribed around the base, which is 6 cm. Now, what does that tell us? Imagine drawing a circle around the square base so that all four corners of the square touch the circle. The radius of this circle is the distance from the center of the square to any of its corners. This is significant because it’s half the length of the diagonal of the square base. Why is that important? Because if we know half the diagonal, we can easily find the full diagonal of the square base. And once we have the base diagonal, we can use the Pythagorean Theorem to find the side length of the square base. See how it's all starting to connect? Each piece of information unlocks another, bringing us closer to our goal.

By carefully analyzing the given information – the prism's diagonal and the radius of the circumscribed circle – we're setting ourselves up for success. We're identifying the key relationships and geometric properties that will lead us to the length of the lateral edge. Remember, geometry is all about seeing the connections and using the right tools to solve the puzzle.

Applying the Pythagorean Theorem

Okay, guys, now we get to use one of the most famous theorems in all of mathematics: the Pythagorean Theorem. This is our secret weapon when dealing with right triangles, and as we've already figured out, our prism is full of them! The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In simple terms: a² + b² = c². Where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.

So, how does this apply to our prism? Remember that space diagonal we talked about? It forms a right triangle with the diagonal of the base and the lateral edge of the prism. Let's call the length of the lateral edge (which is what we want to find) 'h' (for height), the diagonal of the square base 'd', and the space diagonal (given as 13 cm) 'D'. According to the Pythagorean Theorem in 3D, we have: h² + d² = D². This is a super important equation for us. We know D (it's 13 cm), and we need to find 'h'. To do that, we first need to find 'd', the diagonal of the square base.

Remember that radius of the circumscribed circle (6 cm)? It's half the diagonal of the square base. So, the full diagonal 'd' is twice the radius, which is 2 * 6 cm = 12 cm. Awesome! Now we have 'd'. Let's plug our values into the 3D Pythagorean equation: h² + (12 cm)² = (13 cm)². This simplifies to h² + 144 cm² = 169 cm². Now, we just need to solve for h². Subtracting 144 cm² from both sides gives us h² = 25 cm². Taking the square root of both sides, we get h = 5 cm. Voila! We've found the length of the lateral edge.

The Pythagorean Theorem was our key to unlocking this problem. By carefully identifying the right triangles within the prism and applying the theorem, we were able to connect the given information and solve for the unknown. It’s a fantastic example of how a fundamental mathematical principle can be used to solve complex geometric problems.

Demonstrating the Lateral Edge Length

Alright guys, let’s recap what we've done and formally show that the lateral edge of the prism is indeed 5 cm. We started with a right regular quadrangular prism, a space diagonal of 13 cm, and a circumscribed circle radius of 6 cm. Our goal was to find the length of the lateral edge. We knew we had to use all the pieces of information provided to solve for the unknown, using geometric principles along the way.

First, we recognized that the space diagonal forms a right triangle with the diagonal of the square base and the lateral edge. This allowed us to use the 3D version of the Pythagorean Theorem: h² + d² = D², where 'h' is the lateral edge, 'd' is the base diagonal, and 'D' is the space diagonal. Next, we used the information about the circumscribed circle. Since the radius of the circle is 6 cm, the diagonal of the square base is twice that, which is 12 cm. We now had the value for 'd'.

We then plugged our known values (D = 13 cm and d = 12 cm) into the Pythagorean equation: h² + (12 cm)² = (13 cm)². This simplified to h² + 144 cm² = 169 cm². Solving for h², we got h² = 25 cm². Taking the square root of both sides, we found h = 5 cm. And that's it! We've successfully demonstrated that the lateral edge of the prism is 5 cm.

Therefore, we have shown that the lateral edge of the prism is equal to 5 cm, as required by the problem. This whole process demonstrates the power of combining geometric understanding with algebraic manipulation. We used the properties of the prism, the relationship between the circumscribed circle and the square base, and, of course, the mighty Pythagorean Theorem to reach our solution. Well done, guys!

Conclusion: Mastering Geometry with Prisms

So there you have it, guys! We've successfully navigated the world of right regular quadrangular prisms and shown that the lateral edge is 5 cm. This problem wasn’t just about plugging in numbers; it was about understanding the geometry, visualizing the relationships, and strategically applying the right tools. We saw how the properties of the prism, the circumscribed circle, and the Pythagorean Theorem all came together to solve the puzzle.

The key takeaway here is that geometry is more than just memorizing formulas. It’s about developing spatial reasoning, seeing the connections between different elements, and using logic to break down complex problems into manageable steps. By carefully analyzing the given information, identifying relevant geometric principles, and applying algebraic techniques, we can tackle even the trickiest geometric challenges.

Remember, practice makes perfect! The more you work through these types of problems, the more comfortable you’ll become with the concepts and the problem-solving process. So, keep exploring, keep questioning, and keep building your geometric intuition. Geometry is a fascinating and rewarding field, and with a little effort, you can master it. Keep up the great work, guys, and I'll see you in the next mathematical adventure!