Question

current attempt in progress
solve the triangle.
86.1°
4.00 ft
c
40.5°
a
b

a = 43.4°, b = 3.78 ft, c = 5.81 ft
a = 43.4°, b = 5.81 ft, c = 3.78 ft
a = 53.4°, b = 3.24 ft, c = 4.97 ft
a = 53.4°, b = 4.97 ft, c = 3.24 ft
no triangle with the given measurements.

Explanation:

Step1: Find angle \( \alpha \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( \alpha=180^{\circ}-(40.5^{\circ} + 86.1^{\circ})=180^{\circ}-126.6^{\circ}=53.4^{\circ} \)

Step2: Use the Law of Sines to find \( b \)

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). Here \( a = 4.00\) ft, \(A = 86.1^{\circ}\), \(B=40.5^{\circ}\). Then \( b=\frac{4.00\times\sin(40.5^{\circ})}{\sin(86.1^{\circ})}\). Since \( \sin(40.5^{\circ})\approx0.6494\), \( \sin(86.1^{\circ})\approx0.9970\), \( b=\frac{4.00\times0.6494}{0.9970}\approx2.60/0.9970\approx4.97\) ft

Step3: Use the Law of Sines to find \( c \)

Using \( c=\frac{4.00\times\sin(53.4^{\circ})}{\sin(86.1^{\circ})}\). Since \( \sin(53.4^{\circ})\approx0.8021\), \( c=\frac{4.00\times0.8021}{0.9970}\approx3.21/0.9970\approx3.24\) ft

Answer:

\( \alpha = 53.4^{\circ},b = 4.97\) ft, \(c = 3.24\) ft (the fourth option)