Calculating Side BC: A Triangle Problem Solved

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Calculating Side BC in Triangle ABC: A Comprehensive Guide

Hey guys! Let's dive into a classic geometry problem where we need to find the length of a side in a triangle. This isn't just about crunching numbers; it's about understanding the relationships within shapes. We'll take it step-by-step, so even if math isn't your favorite subject, you'll come out feeling like a geometry pro. So, let’s get started and solve this problem together!

Understanding the Problem

So, here's the scenario: we have a triangle, which we'll call ABC. We know a few things about it:

  • The angle at vertex A (angle CAB) is 60 degrees.
  • The side AB is 10 centimeters long.
  • The side AC is 8 centimeters long.

Our mission, should we choose to accept it (and we do!), is to find out the length of the side BC. This is a pretty common type of problem in geometry, and the key to cracking it is knowing the right tools and how to use them. We're not just blindly applying formulas here; we're thinking strategically about how the pieces of the triangle fit together.

Visualizing the Triangle

Before we jump into calculations, it's super helpful to visualize the triangle. Imagine triangle ABC. Side AB is like the base, side AC is leaning a bit to the right, and the angle between them at A is 60 degrees. We're trying to find the length of the side that closes the triangle, side BC. Drawing a quick sketch can make the relationships between the sides and angles much clearer. When you visualize it, you start to see how the given information connects and what tools might be useful.

Identifying the Right Tool: The Law of Cosines

Okay, so we have two sides and an included angle (the angle between them). This screams for the Law of Cosines! The Law of Cosines is like the Swiss Army knife of triangle problems. It lets you find a missing side if you know two sides and the angle between them. It's a powerful formula that connects the sides and angles in any triangle, not just right triangles. Unlike the Pythagorean theorem, which only works for right triangles, the Law of Cosines is a more general rule.

The Law of Cosines formula looks like this:

c² = a² + b² - 2ab * cos(C)

Where:

  • c is the side we're trying to find (BC in our case).
  • a and b are the other two sides (AB and AC).
  • C is the angle opposite side c (angle CAB).

Now, let’s break down why this formula is so perfect for our problem. We know 'a' (AB = 10 cm), we know 'b' (AC = 8 cm), and we know the angle C (angle CAB = 60 degrees). The only unknown is 'c', which is exactly what we want to find! The Law of Cosines essentially bridges the gap between the sides and the angle, allowing us to calculate the missing side length.

Applying the Law of Cosines

Alright, let's get our hands dirty and actually use the Law of Cosines to solve for the length of BC. This is where the magic happens, where we plug in the numbers and see the solution start to emerge. Don't worry, we'll take it slow and break down each step.

Plugging in the Values

First things first, let's substitute the values we know into the Law of Cosines formula:

BC² = AB² + AC² - 2 * AB * AC * cos(CAB)

Replacing the side names with their lengths and the angle with its measure, we get:

BC² = 10² + 8² - 2 * 10 * 8 * cos(60°)

See how we're just replacing the letters with the numbers we know? This is the key to solving any math problem: break it down into smaller, manageable steps. Now we have a clear equation with just one unknown, BC, which is exactly what we want.

Calculating the Cosine

Before we can simplify further, we need to figure out the value of cos(60°). This is where our knowledge of trigonometry comes in handy. Remember the unit circle or the special right triangles? Cos(60°) is a common trigonometric value, and it's equal to 1/2. You might have memorized this, or you might need to quickly sketch a 30-60-90 triangle to recall the ratios.

So, let's replace cos(60°) with 1/2 in our equation:

BC² = 10² + 8² - 2 * 10 * 8 * (1/2)

Now the equation looks even simpler, and we're one step closer to finding BC!

Simplifying the Equation

Now comes the fun part: simplifying the equation. We'll follow the order of operations (PEMDAS/BODMAS) to make sure we do everything in the right order.

First, let's calculate the squares:

BC² = 100 + 64 - 2 * 10 * 8 * (1/2)

Next, let's take care of the multiplication:

BC² = 100 + 64 - 80

Now, it's just addition and subtraction:

BC² = 164 - 80

BC² = 84

We're almost there! We've found BC squared, but we want BC itself.

Finding the Length of BC

Okay, we've got BC² = 84. To find BC, we need to take the square root of both sides of the equation. Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. This is the final step in unlocking the value of BC.

Taking the Square Root

So, let's take the square root of both sides:

√(BC²) = √84

This simplifies to:

BC = √84

Now, we need to figure out what the square root of 84 is. It's not a perfect square, so we won't get a whole number. We can use a calculator to get an approximate value, or we can simplify the square root by factoring out perfect squares.

Simplifying the Square Root (Optional)

To simplify √84, we look for perfect square factors of 84. We can write 84 as 4 * 21, and 4 is a perfect square (2 * 2 = 4). So:

√84 = √(4 * 21)

We can separate the square root:

√84 = √4 * √21

And simplify the square root of 4:

√84 = 2√21

So, the exact length of BC is 2√21 centimeters. This is a more precise answer than a decimal approximation.

Approximate Value (Using a Calculator)

If we want a decimal approximation, we can use a calculator to find the square root of 84:

√84 ≈ 9.165

So, BC is approximately 9.165 centimeters.

Conclusion

And there you have it! We've successfully found the length of side BC in triangle ABC. By using the Law of Cosines, we were able to relate the sides and angles of the triangle and solve for the unknown side. Whether you leave the answer as 2√21 cm or approximate it to 9.165 cm, you've cracked the code!

Key Takeaways

  • Visualize the problem: A sketch can make all the difference.
  • Choose the right tool: The Law of Cosines is your friend when you have two sides and an included angle.
  • Break it down: Simplify the equation step-by-step.
  • Don't be afraid of square roots: They're just the inverse of squaring a number.

Geometry problems can seem daunting at first, but with the right approach, you can conquer them. Keep practicing, and you'll become a triangle-solving master in no time! Remember, it's not just about getting the right answer; it's about understanding the process and building your problem-solving skills. Great job, everyone, on tackling this geometry challenge! Now go forth and conquer more triangles!