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


To find the length of side \( g \) in \( \Delta FGH \), we can use the **Law of Sines**. Here's the information provided:

- \( h = 840 \, \text{inches} \) (side opposite angle \( H \)),
- \( m\angle F = 93^\circ \),
- \( m\angle G = 49^\circ \).

### Step 1: Find \( m\angle H \)
The sum of the angles in any triangle is \( 180^\circ \). So, we can calculate \( m\angle H \) as follows:

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

### Step 2: Apply the Law of Sines
The Law of Sines states:

\[
\frac{h}{\sin(m\angle H)} = \frac{g}{\sin(m\angle G)} = \frac{f}{\sin(m\angle F)}
\]

We want to find \( g \), so we use the relationship:

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

Solving for \( g \):

\[
g = \frac{h \cdot \sin(m\angle G)}{\sin(m\angle H)}
\]

### Step 3: Substitute the known values
Using a calculator to find the sine of the angles:

\[
\sin(49^\circ) \approx 0.7547, \quad \sin(38^\circ) \approx 0.6157
\]

Now, substitute the values into the equation:

\[
g = \frac{840 \times 0.7547}{0.6157} \approx \frac{634.0}{0.6157} \approx 1029.8
\]

### Final Answer:
The length of side \( g \) is approximately **1029.8 inches**.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the Law of Sines be applied if one of the angles was obtuse?
2. What is the relationship between the angles and sides in a triangle?
3. How can we determine if a triangle is possible given certain angles and side lengths?
4. How would the Law of Cosines be applied if two sides and an included angle were known?
5. How does the area of a triangle relate to the sine of its angles?

**Tip:** Always double-check angle sums in a triangle—they must add up to 180°.

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

Learn how to find the length of side g in ΔFGH using the Law of Sines. Given h = 840 inches, m∠F = 93°, and m∠G = 49°, this problem provides a step-by-step trigonometric solution suitable for Grades 10-12.

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