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We are given a triangle \(\triangle FGH\) where the following information is provided:

- \(h = 840\) inches (side opposite angle \(G\)),
- \(\angle F = 93^\circ\),
- \(\angle G = 49^\circ\).

We need to find the length of side \(g\), to the nearest tenth of an inch.

### Step 1: Find the third angle of the triangle

The sum of the angles in a triangle is \(180^\circ\). So we can calculate \(\angle H\):

\[
\angle H = 180^\circ - \angle F - \angle G = 180^\circ - 93^\circ - 49^\circ = 38^\circ
\]

### Step 2: Use the Law of Sines

The Law of Sines states:

\[
\frac{h}{\sin \angle G} = \frac{g}{\sin \angle H}
\]

Substituting the known values:

\[
\frac{840}{\sin 49^\circ} = \frac{g}{\sin 38^\circ}
\]

### Step 3: Solve for \(g\)

\[
g = \frac{840 \cdot \sin 38^\circ}{\sin 49^\circ}
\]

Using a calculator:

\[
g = \frac{840 \cdot 0.6157}{0.7547} \approx 684.91
\]

Thus, the length of side \(g\) is approximately \(684.9\) inches to the nearest tenth.

Would you like more details or further clarification?

Here are 5 related questions to explore:

1. What is the Law of Sines, and when can it be applied?
2. How does knowing the sum of angles in a triangle help in solving problems?
3. What are some real-world applications of the Law of Sines?
4. How would the result change if one of the angles was slightly different?
5. Can the Law of Cosines be used to solve this problem?

**Tip:** When using the Law of Sines, make sure your calculator is set to the correct mode (degrees or radians) depending on the angle units.

In ΔFGH, h = 840 inches, ∠F = 93° and ∠G = 49°. Find the length of g, to the nearest 10th of an inch.

Learn how to find the side length of a triangle using the Law of Sines. In this example, ΔFGH has h = 840 inches, ∠F = 93°, and ∠G = 49°. Using the Law of Sines, you can calculate the unknown side g to the nearest tenth of an inch.

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