To determine the number of triangles that can be formed with the given information, we can use the Law of Sines and the properties of triangles.
Given:
- Angle \( A = 48^\circ \)
- Side \( a = 10 , m \) (the side opposite angle \( A \))
- Side \( b = 12 , m \) (the side opposite angle \( B \))
We apply the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Substituting the known values:
\[ \frac{10}{\sin 48^\circ} = \frac{12}{\sin B} \]
Now we can isolate \( \sin B \):
\[ \sin B = \frac{12 \cdot \sin 48^\circ}{10} \]
Calculating \( \sin 48^\circ \):
\[ \sin 48^\circ \approx 0.7431 \]
Now, substituting this value back into our equation:
\[ \sin B \approx \frac{12 \cdot 0.7431}{10} \approx \frac{8.9172}{10} \approx 0.89172 \]
Next, we check if \( \sin B \) is valid. The range for sine values is between \( -1 \) and \( 1 \), so \( \sin B \approx 0.89172 \) is a viable value.
Now, we can find angle \( B \):
\[ B = \sin^{-1}(0.89172) \]
Calculating \( B \):
\[ B \approx 63.6^\circ \]
Since \( B \) is a valid angle, we should also consider the possibility of a second triangle. The second triangle occurs when we have:
\[ B' = 180^\circ - B \approx 180^\circ - 63.6^\circ = 116.4^\circ \]
Next, we check if angle \( C \) for the second triangle is valid by ensuring the angles sum up to less than \( 180^\circ \):
For the first triangle:
\[ C = 180^\circ - A - B \approx 180^\circ - 48^\circ - 63.6^\circ \approx 68.4^\circ \]
This angle is valid.
For the second triangle:
\[ C' = 180^\circ - A - B' \approx 180^\circ - 48^\circ - 116.4^\circ \approx 15.6^\circ \]
This angle is also valid.
Since both triangles are valid:
Conclusion: There are two triangles for the given information.