Given the triangle
50
x
20
30
[Graphs generated by this script: initPicture(-15,55,-15,55);line([0,0],[25,50]);line([25,50],[50,0]);line([0,0],[50,0]);text([5,25],"50");text([45,25],"x");text([38,4],"20");text([8,4],"30");circle([13,7],1);circle([43,7],1)], find the length of side
x
using the Law of Sines. Round your final answer to 2 decimal places.

x
=

To solve for the length of side \( x \) in the given triangle using the Law of Sines, let's break down the problem step by step.

### Step 1: Identify the given values and the Law of Sines
In the problem, you have:
- Side \( a = 20 \)
- Side \( b = 30 \)
- Angle opposite side \( a = 50^\circ \)
- Angle opposite side \( b = x \)

The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

### Step 2: Apply the Law of Sines
Using the Law of Sines, we can write:
\[
\frac{20}{\sin(50^\circ)} = \frac{x}{\sin(30^\circ)}
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \):
\[
x = \frac{20 \times \sin(30^\circ)}{\sin(50^\circ)}
\]

Given:
\[
\sin(30^\circ) = 0.5 \quad \text{and} \quad \sin(50^\circ) \approx 0.7660
\]

Substitute the values:
\[
x = \frac{20 \times 0.5}{0.7660} \approx \frac{10}{0.7660} \approx 13.06
\]

### Final Answer
The length of side \( x \) is approximately \( \boxed{13.06} \).

---

Would you like more details on the Law of Sines or have any other questions?

Here are some related questions for further practice:

1. How do you apply the Law of Cosines to find the length of a side in a triangle?
2. How can you find the third angle in a triangle if two angles are known?
3. What is the difference between the Law of Sines and the Law of Cosines?
4. How would you solve a triangle if all three sides are given?
5. How do you determine if a triangle is obtuse using side lengths?
6. What are the conditions for using the Law of Sines?
7. Can you use the Law of Sines for non-right triangles? How?
8. How would the solution change if the given angle were larger than 90 degrees?

**Tip:** When using trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) based on the angle measurements given.

Learn how to find the length of side x in a triangle using the Law of Sines with a step-by-step solution. This problem involves trigonometric concepts, including the Law of Sines formula. Suitable for high school students.

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