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To find the area of this triangle, we can use the formula for the area of a triangle given two sides and the included angle:

\[
\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)
\]

where:
- \( a = 7 \)
- \( b = 9 \)
- \( C = \frac{5\pi}{6} \)

Let's go through the steps:

1. **Calculate the Sine of the Angle**:
   - Since \( \frac{5\pi}{6} \) radians is equivalent to \( 150^\circ \),
   \[
   \sin\left(\frac{5\pi}{6}\right) = \sin(150^\circ) = 0.5
   \]

2. **Plug in the Values**:
   \[
   \text{Area} = \frac{1}{2} \times 7 \times 9 \times 0.5
   \]

3. **Calculate**:
   \[
   \text{Area} = \frac{1}{2} \times 63 \times 0.5 = 15.75
   \]

So, the area of the triangle is **15.75 square units**, rounded to the nearest hundredth.

Would you like further explanation on any of these steps?

Here are some related questions for additional practice:
1. How would you find the area if only one side and the height were given?
2. What is the area of a triangle with sides 8 and 10, and an included angle of \(45^\circ\)?
3. How does changing the angle affect the area of a triangle with fixed side lengths?
4. What is the relationship between the sine of an angle and its supplementary angle?
5. How would the formula change if the triangle were a right triangle?

**Tip:** When working with radians, it’s helpful to memorize the common sine values for angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{5\pi}{6} \).

Find the area of a triangle with sides 7 and 9, and an included angle of 5π/6.

Learn how to find the area of a triangle with side lengths 7 and 9, and an included angle of 5π/6 radians. This problem demonstrates the use of trigonometry in triangle area calculation, including the formula Area = 0.5 * a * b * sin(C). Suitable for Grades 9-10 students.

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