To find the area of a triangle when you have two sides and the included angle, you can use the formula:
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]
where \(a\) and \(b\) are the lengths of the two sides, and \(C\) is the included angle.
In this case:
- \(a = 8 , \text{cm}\)
- \(b = 12 , \text{cm}\)
- \(C = 60^\circ\)
Now, we can plug these values into the formula:
\[ \text{Area} = \frac{1}{2} \times 8 \times 12 \times \sin(60^\circ) \]
The value of \(\sin(60^\circ)\) is \(\frac{\sqrt{3}}{2}\).
Now substituting in:
\[ \text{Area} = \frac{1}{2} \times 8 \times 12 \times \frac{\sqrt{3}}{2} \] \[ = 48 \times \frac{\sqrt{3}}{2} \] \[ = 24\sqrt{3} \]
Calculating \(24\sqrt{3}\):
\[ \approx 24 \times 1.732 \approx 41.57 , \text{cm}^2 \]
Thus, the area of the triangle is approximately \(41.57 , \text{cm}^2\).
The correct response is: 41.57 cm².