To solve this triangle, we can use the Law of Sines since we know:

1. Angle $B = 41^\circ$
2. Angle $C = 28^\circ$
3. Side $b = 25$

We are looking to find the measurements of:

• Angle $A$
• Sides $a$ and $c$

Step 1: Find Angle $A$

Since the sum of the angles in any triangle is $180^\circ$, we can calculate $A$ as follows:

$A = 180^\circ - B - C = 180^\circ - 41^\circ - 28^\circ = 111^\circ$

Step 2: Use the Law of Sines to Find Side $a$

The Law of Sines states:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

Rearrange to solve for $a$:

$a = \frac{b \cdot \sin(A)}{\sin(B)}$

Substitute the known values:

$a = \frac{25 \cdot \sin(111^\circ)}{\sin(41^\circ)}$

Calculating the sine values and solving for $a$:

1. $\sin(111^\circ) \approx 0.8910$
2. $\sin(41^\circ) \approx 0.6561$

$a \approx \frac{25 \cdot 0.8910}{0.6561} \approx 33.9$

Step 3: Use the Law of Sines to Find Side $c$

Similarly, solve for $c$:

$c = \frac{b \cdot \sin(C)}{\sin(B)}$

Substitute the known values:

$c = \frac{25 \cdot \sin(28^\circ)}{\sin(41^\circ)}$

Calculating the sine values:

1. $\sin(28^\circ) \approx 0.4695$
2. $\sin(41^\circ) \approx 0.6561$

$c \approx \frac{25 \cdot 0.4695}{0.6561} \approx 17.9$

Conclusion

The solution for this triangle is:

• $A \approx 111^\circ$
• $a \approx 34$ (rounded to the nearest integer)
• $c \approx 18$ (rounded to the nearest integer)

Given these calculations, the correct answer choice would be B with:

• $A \approx 111^\circ$
• $a \approx 34$
• $c \approx 18$

Do you have any questions or would like further explanation on any step?

Here are 5 related questions for further practice:

1. How does the Law of Sines work for finding unknown angles in a triangle?
2. When is it possible to have two solutions in a triangle using the Law of Sines?
3. How would you solve for unknown sides if you were given two sides and an angle not between them?
4. What is the difference between using the Law of Sines and the Law of Cosines?
5. How can rounding affect the accuracy of triangle solutions?

Tip: When using the Law of Sines, always check if an ambiguous case (SSA) exists, which could lead to two possible solutions.