https://file.aiquickdraw.com/mathbot/file/17290934269603ugf2zar.jpg

To find the area of a triangle when two sides and the included angle are given, we use the **formula for the area** of a triangle:

\[
\text{Area} = \frac{1}{2}ab \sin(C)
\]

Where:
- \(a\) and \(b\) are the two sides,
- \(C\) is the included angle between them,
- \(\sin(C)\) is the sine of the angle in degrees.

Let’s go step by step for each triangle in the image.

### 1. Triangle 1:
- Side \(a = 9 \, \text{yd}\)
- Side \(b = 11 \, \text{yd}\)
- Included angle \(C = 46^\circ\)

\[
\text{Area} = \frac{1}{2} \times 9 \times 11 \times \sin(46^\circ)
\]
\[
\text{Area} = \frac{1}{2} \times 99 \times 0.7193 \approx 35.6 \, \text{square yards}
\]

### 2. Triangle 2:
- Side \(a = 12 \, \text{m}\)
- Side \(b = 15 \, \text{m}\)
- Included angle \(C = 58^\circ\)

\[
\text{Area} = \frac{1}{2} \times 12 \times 15 \times \sin(58^\circ)
\]
\[
\text{Area} = \frac{1}{2} \times 180 \times 0.848 \approx 76.3 \, \text{square meters}
\]

### 3. Triangle 3:
- Side \(a = 9 \, \text{cm}\)
- Side \(b = 9 \, \text{cm}\)
- Included angle \(C = 40^\circ\)

\[
\text{Area} = \frac{1}{2} \times 9 \times 9 \times \sin(40^\circ)
\]
\[
\text{Area} = \frac{1}{2} \times 81 \times 0.6428 \approx 26.0 \, \text{square centimeters}
\]

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Let me know if you need further explanation or more details on any of the steps. 

Here are 5 related questions you might consider:
1. How does the sine rule apply in non-right angled triangles?
2. What is the Law of Cosines, and how does it differ from the formula used here?
3. How can we derive the area of a triangle without knowing the angle?
4. How do you calculate the sine of an angle manually?
5. What other methods are there to find the area of a triangle?

**Tip:** Remember that the sine of an angle is always a ratio between the length of the opposite side and the hypotenuse in a right triangle.

Find the area of triangles ABC to the nearest tenth using the given side lengths and included angles.

Learn how to calculate the area of a triangle using the sine rule. Given side lengths and an included angle, find the area for three triangles: one with sides 9 yd and 11 yd, angle 46°, one with sides 12 m and 15 m, angle 58°, and one with sides 9 cm, angle 40°. Suitable for high school geometry students.

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