Finding The Circumradius Of Triangle ABC A Step-by-Step Guide

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Hey guys! Geometry can sometimes feel like navigating a maze, but with the right tools and understanding, we can solve even the trickiest problems. Today, we're going to tackle a classic geometry question: finding the radius of the circumcircle of a triangle. Let's dive into a problem involving triangle ABC with sides AB = 4 cm, BC = 5 cm, and AC = 6 cm. Our mission? To find the radius of the circle that perfectly circumscribes this triangle. We'll walk through the steps together, making sure everything is crystal clear. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand the problem inside and out. Understanding the circumcircle is crucial. Imagine a circle drawn around a triangle such that all three vertices of the triangle lie perfectly on the circle's circumference. This circle is called the circumcircle, and its center is the circumcenter. The radius of this circle is what we're trying to find – the circumradius.

Why is this important? Well, the circumradius gives us valuable information about the triangle's properties and its relationship to circles. It's a concept that pops up in various areas of geometry, from triangle constructions to advanced theorems. So, mastering this concept is definitely worth our time. The circumradius, often denoted by R, is directly related to the triangle's sides and its area. This relationship is key to solving our problem. We'll be using a formula that connects the circumradius to the sides of the triangle and its area. So, let's keep this in mind as we move forward.

Now, let's think about the information we already have. We know the lengths of all three sides of triangle ABC: AB = 4 cm, BC = 5 cm, and AC = 6 cm. This is a great starting point! Knowing all three sides allows us to calculate the area of the triangle using Heron's formula, which we'll discuss in detail later. Once we have the area, we can plug the values into the circumradius formula and find our answer. This problem isn't just about plugging numbers into a formula; it's about understanding the relationships between different geometric concepts. By understanding the circumcircle, the circumradius, and how they relate to the triangle's sides and area, we're building a strong foundation in geometry. So, let's move on to the next step: calculating the area of the triangle. This will bring us one step closer to finding the circumradius.

Calculating the Area Using Heron's Formula

Alright, let's get our hands dirty with some calculations! The first key step in finding the circumradius is to determine the area of triangle ABC. And for this, we'll use a nifty tool called Heron's formula. Heron's formula is a gem in geometry because it allows us to calculate the area of a triangle using only the lengths of its sides. No angles needed! This is perfect for our situation since we know AB = 4 cm, BC = 5 cm, and AC = 6 cm.

So, what exactly is Heron's formula? It looks a bit intimidating at first, but don't worry, we'll break it down. The formula states that the area (A) of a triangle with sides a, b, and c is given by:

A = √[s(s - a)(s - b)(s - c)]

Where 's' is the semi-perimeter of the triangle, which is simply half of the triangle's perimeter. In other words:

s = (a + b + c) / 2

Now, let's apply this to our triangle ABC. First, we need to calculate the semi-perimeter (s). In our case, a = 4 cm, b = 5 cm, and c = 6 cm. So:

s = (4 + 5 + 6) / 2 = 15 / 2 = 7.5 cm

Great! We have the semi-perimeter. Now, we can plug everything into Heron's formula:

A = √[7.5(7.5 - 4)(7.5 - 5)(7.5 - 6)] A = √[7.5 * 3.5 * 2.5 * 1.5] A = √[98.4375] A ≈ 9.92 cm²

So, the area of triangle ABC is approximately 9.92 square centimeters. Now that we have the area, we're one step closer to finding the circumradius. Remember, the circumradius formula connects the sides of the triangle, its area, and the circumradius itself. With the area in hand, we can now use this formula to find the radius of the circumcircle. Calculating the area using Heron's formula might seem like a detour, but it's a crucial step in solving the problem. It highlights the interconnectedness of different geometric concepts and how we can leverage them to find solutions. So, let's move on and use this area to find the circumradius!

Applying the Circumradius Formula

Okay, we've successfully calculated the area of triangle ABC using Heron's formula. Now comes the exciting part – using this area to find the circumradius! The circumradius formula is the bridge that connects the sides of a triangle, its area, and the radius of its circumcircle. It's a powerful tool in our geometry arsenal, and it's exactly what we need to solve this problem.

The formula itself looks like this:

R = (a * b * c) / (4 * A)

Where:

  • R is the circumradius (what we're trying to find).
  • a, b, and c are the lengths of the sides of the triangle.
  • A is the area of the triangle.

Notice how the formula elegantly combines the side lengths and the area to give us the circumradius. This formula is derived from the Law of Sines and the relationship between the area of a triangle and its circumradius. It's a beautiful example of how different geometric concepts intertwine.

Now, let's plug in the values we have for triangle ABC. We know that a = 4 cm, b = 5 cm, c = 6 cm, and we calculated the area (A) to be approximately 9.92 cm². Substituting these values into the formula, we get:

R = (4 * 5 * 6) / (4 * 9.92) R = 120 / 39.68 R ≈ 3.02 cm

So, the circumradius of triangle ABC is approximately 3.02 cm. This means that the radius of the circle that perfectly circumscribes triangle ABC is about 3.02 centimeters. We've successfully used the circumradius formula to find our answer! Remember, this formula is a powerful tool for solving problems involving triangles and their circumcircles. It allows us to connect the sides, area, and circumradius in a single equation. By understanding and applying this formula, we can tackle a wide range of geometry problems. Now, let's take a look at the answer choices provided and see which one matches our calculated value.

Comparing with the Answer Choices and Final Answer

We've done the hard work of calculating the circumradius of triangle ABC. We found that R ≈ 3.02 cm. Now, it's time to compare our result with the answer choices given in the problem. This is a crucial step because it allows us to verify our calculations and ensure we've arrived at the correct solution. The answer choices provided are:

  1. rac{3\sqrt{7}}{8}
  2. rac{8\sqrt{7}}{7}
  3. rac{8 ext{\sqrt{3}}}{3}
  4. rac{18 ext{\sqrt{5}}}{5}
  5. 4.5

To accurately compare our result with these options, we need to convert the radical expressions into decimal approximations. Let's calculate the approximate values for the first two options:

  1. rac{3 ext{\sqrt{7}}}{8} ≈ (3 * 2.646) / 8 ≈ 0.992
  2. rac{8 ext{\sqrt{7}}}{7} ≈ (8 * 2.646) / 7 ≈ 3.024

Ah ha! It looks like option 2, rac{8 ext{\sqrt{7}}}{7}, is a very close match to our calculated circumradius of approximately 3.02 cm. The other options are significantly different from our result, so we can confidently rule them out.

Therefore, the correct answer is 2) rac{8 ext{\sqrt{7}}}{7}. We've successfully found the circumradius of triangle ABC by using Heron's formula to calculate the area and then applying the circumradius formula. We also carefully compared our result with the answer choices to ensure accuracy. This problem demonstrates the power of combining different geometric concepts and formulas to solve complex problems. We started by understanding the problem, then used Heron's formula, applied the circumradius formula, and finally, compared our result with the answer choices. This step-by-step approach is key to success in geometry. Remember, geometry is all about understanding relationships and using the right tools to uncover hidden truths. By mastering these concepts and techniques, we can confidently tackle any geometry challenge that comes our way.

So, guys, we've successfully navigated the world of circumcircles and circumradius! We started with a triangle, used Heron's formula to find its area, and then applied the circumradius formula to determine the radius of the circle that perfectly surrounds it. And after comparing our result with the given options, we confidently arrived at the answer: 877\frac{8\sqrt{7}}{7}. This journey through geometry highlights the importance of understanding key concepts, applying the right formulas, and taking a systematic approach to problem-solving. Remember, geometry isn't just about memorizing formulas; it's about understanding the relationships between shapes, lines, and angles. By building a strong foundation in these concepts, we can unlock the beauty and power of geometry. Keep practicing, keep exploring, and keep challenging yourselves with new problems. Geometry is a fascinating field, and with a little effort, you can master it! And remember, every problem solved is a step closer to geometric mastery. So, let's keep learning and keep growing our geometric skills!