Solve the triangle.
aequals4.45 m    cequals5.55 m    Bequals26.7degrees

We are given the following information about the triangle:

- \( a = 4.45 \, \text{m} \)
- \( c = 5.55 \, \text{m} \)
- \( B = 26.7^\circ \)

To solve the triangle, we need to find the unknown side \( b \) and the unknown angles \( A \) and \( C \).

### Step 1: Use the Law of Sines to find angle \( C \)
The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

First, let's find angle \( C \). Since the sum of the angles in a triangle is \( 180^\circ \), we can find angle \( C \) as follows:

\[
C = 180^\circ - B - A
\]

But we need to find angle \( A \) first.

### Step 2: Use the Law of Cosines to find side \( b \)
The Law of Cosines relates the sides and angles of a triangle:

\[
b^2 = a^2 + c^2 - 2ac \cdot \cos B
\]

Substitute the values for \( a \), \( c \), and \( B \):

\[
b^2 = 4.45^2 + 5.55^2 - 2 \cdot 4.45 \cdot 5.55 \cdot \cos(26.7^\circ)
\]

Calculate:

\[
b^2 = 19.8025 + 30.8025 - 2 \cdot 4.45 \cdot 5.55 \cdot 0.894
\]
\[
b^2 = 19.8025 + 30.8025 - 44.26194
\]
\[
b^2 = 50.605 - 44.26194
\]
\[
b^2 = 6.34306
\]
\[
b = \sqrt{6.34306} \approx 2.52 \, \text{m}
\]

### Step 3: Use the Law of Sines to find angle \( A \)
Now that we know \( b \), we can use the Law of Sines to find \( A \):

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
\[
\frac{4.45}{\sin A} = \frac{2.52}{\sin 26.7^\circ}
\]

Solve for \( \sin A \):

\[
\sin A = \frac{4.45 \cdot \sin 26.7^\circ}{2.52}
\]
\[
\sin A = \frac{4.45 \cdot 0.4488}{2.52}
\]
\[
\sin A = \frac{1.996}{2.52} \approx 0.7917
\]

Now find \( A \):

\[
A = \arcsin(0.7917) \approx 52.2^\circ
\]

### Step 4: Find angle \( C \)
Now that we know \( A \) and \( B \), we can find angle \( C \):

\[
C = 180^\circ - 52.2^\circ - 26.7^\circ
\]
\[
C = 101.1^\circ
\]

### Final Answer:
- \( A \approx 52.2^\circ \)
- \( B = 26.7^\circ \) (given)
- \( C \approx 101.1^\circ \)
- \( b \approx 2.52 \, \text{m} \)

Let me know if you would like further details or explanations!

Here are 5 questions to think about:
1. How does the Law of Cosines differ from the Law of Sines?
2. What would happen if \( B \) were obtuse?
3. How can we solve a triangle if only two angles and one side are given?
4. How does the sum of the angles affect the shape of the triangle?
5. Can we apply these rules to non-right triangles?

**Tip**: When solving triangles, always ensure the sum of the angles adds up to 180° to check your work!

Solve the triangle. a = 4.45 m, c = 5.55 m, B = 26.7 degrees.

Step-by-step solution for solving a triangle with side lengths a = 4.45 m, c = 5.55 m, and angle B = 26.7°. Learn how to apply the Law of Sines and Law of Cosines to find unknown sides and angles. Ideal for high school students (Grades 9-12).

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