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 $C = 40^\circ$

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

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.