Finding AC In Triangle ABC A Step-by-Step Geometry Guide

Hey guys! Geometry can be tricky, but don't worry, we're going to break down this problem step by step. We've got a triangle ABC where angle A is 45 degrees, angle B is 30 degrees, and side BC is $8\sqrt{2}$. Our mission? To find the length of side AC. Let’s dive in!

Understanding the Problem

Before we jump into calculations, let's visualize what we have. We have a triangle, and we know two angles and one side. This sounds like a perfect scenario for using the Law of Sines. The Law of Sines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. Specifically, it states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. This law is incredibly useful when dealing with triangles where you know angles and side lengths, and you need to find missing information.

The Law of Sines is expressed as:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite those sides.

In our case:

• Angle A = 45°
• Angle B = 30°
• Side BC (which is opposite angle A) = $8\sqrt{2}$
• We need to find side AC (which is opposite angle B).

So, now we have a clear picture of the problem and the tool we’re going to use. Let’s get into the nitty-gritty of solving it!

Step 1: Find Angle C

Okay, first things first, we need to find angle C. Remember, the angles in a triangle always add up to 180 degrees. This is a basic yet crucial concept in geometry. The sum of the interior angles of any triangle, regardless of its shape or size, is always 180 degrees. This property is a cornerstone of triangle geometry and is essential for solving many problems involving triangles. Knowing this, we can easily find the missing angle if we have the other two angles.

So, we can write:

$$A + B + C = 180°$$

We know A = 45° and B = 30°, so let’s plug those in:

$$45° + 30° + C = 180°$$

Now, let's simplify:

$$75° + C = 180°$$

To find C, we subtract 75° from both sides:

$$C = 180° - 75°$$

$$C = 105°$$

Great! We've found angle C, which is 105 degrees. This piece of information is crucial because it completes our angle set and will help us apply the Law of Sines effectively. Now that we have all three angles, we can move on to using the Law of Sines to find the length of side AC. Let’s keep going!

Step 2: Apply the Law of Sines

Alright, now that we know all three angles, we can use the Law of Sines to find the length of AC. Remember, the Law of Sines states:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

In our case:

• a = BC = $8\sqrt{2}$
• A = 45°
• b = AC (This is what we want to find)
• B = 30°

We can set up the equation using the information we have:

$$\frac{8\sqrt{2}}{\sin 45°} = \frac{AC}{\sin 30°}$$

Now, let's plug in the values of the sines. We know that $\sin 45° = \frac{\sqrt{2}}{2}$ and $\sin 30° = \frac{1}{2}$. So, our equation becomes:

$$\frac{8\sqrt{2}}{\frac{\sqrt{2}}{2}} = \frac{AC}{\frac{1}{2}}$$

This looks a bit complex, but don't worry, we'll simplify it step by step. This is where our algebra skills come in handy. By carefully manipulating the equation, we can isolate AC and find its value. Remember, the key is to keep the equation balanced and perform the same operations on both sides. Let’s break it down and make it easier to solve!

Step 3: Simplify the Equation

Okay, let's simplify this equation. We have:

$$\frac{8\sqrt{2}}{\frac{\sqrt{2}}{2}} = \frac{AC}{\frac{1}{2}}$$

First, let’s simplify the left side. Dividing by a fraction is the same as multiplying by its reciprocal, so we get:

$$8\sqrt{2} \cdot \frac{2}{\sqrt{2}} = \frac{AC}{\frac{1}{2}}$$

The $\sqrt{2}$ in the numerator and denominator cancel each other out:

$$8 \cdot 2 = \frac{AC}{\frac{1}{2}}$$

$$16 = \frac{AC}{\frac{1}{2}}$$

Now, to get AC by itself, we multiply both sides by $\frac{1}{2}$:

$$16 \cdot \frac{1}{2} = AC$$

$$8 = AC$$

So, we've found that AC = 8. Awesome! We're almost there. We've simplified the equation and isolated AC, giving us the solution. Now, let's just make sure we've got everything right and present our final answer.

Step 4: State the Final Answer

Alright, guys, we've done it! We found that AC = 8. So, to wrap it up:

The length of side AC in triangle ABC is 8.

Isn't it satisfying when a tricky problem comes together? Geometry can be a puzzle, but with the right steps and a bit of practice, you can totally nail it. Remember, the Law of Sines is your friend when you have angles and sides to work with. Keep practicing, and you'll become a geometry whiz in no time!

Conclusion

So, there you have it! We successfully found the length of side AC in triangle ABC using the Law of Sines. Remember the key steps:

1. Understand the problem: Visualize the triangle and identify what you need to find.
2. Use the Law of Sines: Set up the equation with the given information.
3. Simplify: Use your algebra skills to solve for the unknown side.
4. State the answer: Clearly present your solution.

Geometry might seem intimidating at first, but breaking it down into smaller steps makes it much more manageable. Keep practicing and exploring different problems, and you'll build confidence in your problem-solving abilities. Great job, and happy calculating!