Solve the triangle. $$ \mathrm{a}=4.19 \mathrm{~m} \quad \mathrm{c}=5.52 \mathrm{~m} \quad \mathrm{~m}=28.1^{\circ} $$ What is the length of side b ? $$ \square^{\mathrm{m}} $$ (Simplify your answer. Type an integer or a decimal. Round to four decimal places if needed.) What is the measure of angle A? $$ \square^{\circ} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if needed.) $$ \square^{\circ} $$ (Simplify is the measure of angle C ?

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To solve the triangle with the given values $$ a = 4.19 \, \text{m} $$, $$ c = 5.52 \, \text{m} $$, and $$ m = 28.1^\circ $$ (where $$ m $$ is angle $$ B $$), we can use the Law of Sines and the Law of Cosines.

### Step 1: Find side $$ b $$

Using the Law of Sines:

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

First, we need to find angle $$ A $$ using the Law of Cosines:

$$
c^2 = a^2 + b^2 - 2ab \cos B
$$

However, we need to find $$ b $$ first. We can rearrange the Law of Sines to find $$ b $$:

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

But we need angle $$ A $$ first. We can find angle $$ C $$ using the Law of Sines as well.

### Step 2: Find angle $$ A $$

Using the Law of Sines:

$$
\frac{a}{\sin A} = \frac{c}{\sin C}
$$

We can express $$ \sin C $$ in terms of $$ \sin A $$:

$$
\sin C = \frac{c \cdot \sin A}{a}
$$

Now, we can use the fact that the sum of angles in a triangle is $$ 180^\circ $$:

$$
A + B + C = 180^\circ
$$

Substituting $$ B $$:

$$
A + 28.1^\circ + C = 180^\circ
$$

Thus,

$$
C = 180^\circ - A - 28.1^\circ
$$

### Step 3: Use the Law of Cosines to find angle $$ A $$

We can also use the Law of Cosines directly to find angle $$ A $$:

$$
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
$$

But we need $$ b $$ first. Let's use the Law of Sines to find $$ b $$:

### Step 4: Find angle $$ C $$

Using the Law of Sines:

$$
\frac{c}{\sin C} = \frac{a}{\sin A}
$$

We can find $$ C $$ using the Law of Sines:

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

### Step 5: Calculate $$ b $$

Using the Law of Cosines:

$$
b^2 = a^2 + c^2 - 2ac \cos B
$$

Substituting the known values:

$$
b^2 = (4.19)^2 + (5.52)^2 - 2 \cdot (4.19) \cdot (5.52) \cdot \cos(28.1^\circ)
$$

Calculating each term:

$$
b^2 = 17.5561 + 30.5504 - 2 \cdot 4.19 \cdot 5.52 \cdot \cos(28.1^\circ)
$$

Calculating $$ \cos(28.1^\circ) $$:

$$
\cos(28.1^\circ) \approx 0.8746
$$

Now substituting:

$$
b^2 = 17.5561 + 30.5504 - 2 \cdot 4.19 \cdot 5.52 \cdot 0.8746
$$

Calculating $$ 2 \cdot 4.19 \cdot 5.52 \cdot 0.8746 $$:

$$
2 \cdot 4.19 \cdot 5.52 \cdot 0.8746 \approx 41.052
$$

Now substituting back:

$$
b^2 = 17.5561 + 30.5504 - 41.052
$$

Calculating:

$$
b^2 = 48.1065 - 41.052 \approx 7.0545
$$

Taking the square root:

$$
b \approx \sqrt{7.0545} \approx 2.65 \, \text{m}
$$

### Step 6: Find angle $$ A $$

Now we can find angle $$ A $$ using the Law of Sines:

$$
\frac{a}{\sin A} = \frac{b}{\sin B}
$$

Rearranging gives:

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

Substituting the known values:

$$
\sin A = \frac{4.19 \cdot \sin(28.1^\circ)}{2.65}
$$

Calculating $$ \sin(28.1^\circ) $$:

$$
\sin(28.1^\circ) \approx 0.4695
$$

Now substituting:

$$
\sin A = \frac{4.19 \cdot 0.4695}{2.65} \approx \frac{1.964}{2.65} \approx 0.7415
$$

Now finding angle $$ A $$:

$$
A \approx \arcsin(0.7415) \approx 48.5^\circ
$$

### Step 7: Find angle $$ C $$

Using the angle sum property:

$$
C = 180^\circ - A - B = 180^\circ - 48.5^\circ - 28.1^\circ \approx 103.4^\circ
$$

### Final Answers

- Length of side $$ b $$: $$ \approx 2.6500 \, \text{m} $$
- Measure of angle $$ A $$: $$ \approx 48.5^\circ $$
- Measure of angle $$ C $$: $$ \approx 103.4^\circ $$

Thus, the final answers are:

- $$ b \approx 2.6500 \, \text{m} $$
- $$ A \approx 48.5^\circ $$
- $$ C \approx 103.4^\circ $$
Length of side $$ b $$: $$ \approx 2.6500 \, \text{m} $$ Measure of angle $$ A $$: $$ \approx 48.5^\circ $$ Measure of angle $$ C $$: $$ \approx 103.4^\circ $$

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