Question

determine the remaining sides and angles of the triangle abc.
what is the measure of angle b?
(simplify your answer. type an integer or a decimal.)
what is the length of side a?
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)
what is the length of side b?
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

To solve for the remaining sides and angles of triangle \( ABC \), we use the Law of Sines and the fact that the sum of angles in a triangle is \( 180^\circ \).

Step 1: Find angle \( B \)

The sum of angles in a triangle is \( 180^\circ \). We know \( \angle A = 24.7^\circ \) and \( \angle C = 107.1^\circ \).

$$ \angle B = 180^\circ - \angle A - \angle C $$

$$ \angle B = 180^\circ - 24.7^\circ - 107.1^\circ $$

$$ \angle B = 48.2^\circ $$

Step 2: Find side \( a \) (Law of Sines)

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We know \( c = 55.2 \) ft, \( \angle A = 24.7^\circ \), \( \angle C = 107.1^\circ \).

$$ \frac{a}{\sin 24.7^\circ} = \frac{55.2}{\sin 107.1^\circ} $$

$$ a = \frac{55.2 \cdot \sin 24.7^\circ}{\sin 107.1^\circ} $$

Using a calculator:
\( \sin 24.7^\circ \approx 0.417 \), \( \sin 107.1^\circ \approx 0.954 \)

$$ a = \frac{55.2 \cdot 0.417}{0.954} \approx \frac{22.9184}{0.954} \approx 24.0 \text{ ft} $$

Step 3: Find side \( b \) (Law of Sines)

Using the Law of Sines again:

$$ \frac{b}{\sin B} = \frac{55.2}{\sin 107.1^\circ} $$

$$ b = \frac{55.2 \cdot \sin 48.2^\circ}{\sin 107.1^\circ} $$

Using a calculator:
\( \sin 48.2^\circ \approx 0.745 \)

$$ b = \frac{55.2 \cdot 0.745}{0.954} \approx \frac{41.124}{0.954} \approx 43.1 \text{ ft} $$

Final Answers:

• Measure of angle \( B \): \( \boxed{48.2^\circ} \)
• Length of side \( a \): \( \boxed{24.0} \) ft
• Length of side \( b \): \( \boxed{43.1} \) ft

Answer:

To solve for the remaining sides and angles of triangle \( ABC \), we use the Law of Sines and the fact that the sum of angles in a triangle is \( 180^\circ \).

Step 1: Find angle \( B \)

The sum of angles in a triangle is \( 180^\circ \). We know \( \angle A = 24.7^\circ \) and \( \angle C = 107.1^\circ \).

$$ \angle B = 180^\circ - \angle A - \angle C $$

$$ \angle B = 180^\circ - 24.7^\circ - 107.1^\circ $$

$$ \angle B = 48.2^\circ $$

Step 2: Find side \( a \) (Law of Sines)

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We know \( c = 55.2 \) ft, \( \angle A = 24.7^\circ \), \( \angle C = 107.1^\circ \).

$$ \frac{a}{\sin 24.7^\circ} = \frac{55.2}{\sin 107.1^\circ} $$

$$ a = \frac{55.2 \cdot \sin 24.7^\circ}{\sin 107.1^\circ} $$

Using a calculator:
\( \sin 24.7^\circ \approx 0.417 \), \( \sin 107.1^\circ \approx 0.954 \)

$$ a = \frac{55.2 \cdot 0.417}{0.954} \approx \frac{22.9184}{0.954} \approx 24.0 \text{ ft} $$

Step 3: Find side \( b \) (Law of Sines)

Using the Law of Sines again:

$$ \frac{b}{\sin B} = \frac{55.2}{\sin 107.1^\circ} $$

$$ b = \frac{55.2 \cdot \sin 48.2^\circ}{\sin 107.1^\circ} $$

Using a calculator:
\( \sin 48.2^\circ \approx 0.745 \)

$$ b = \frac{55.2 \cdot 0.745}{0.954} \approx \frac{41.124}{0.954} \approx 43.1 \text{ ft} $$

Final Answers:

• Measure of angle \( B \): \( \boxed{48.2^\circ} \)
• Length of side \( a \): \( \boxed{24.0} \) ft
• Length of side \( b \): \( \boxed{43.1} \) ft