Question

find the area of the triangle abc. a = 49.6° b = 38.9° c = 26.9 m the area of the triangle is □ m² (round to the nearest tenth as needed.)

Explanation:

Step1: Find angle \(C\)

The sum of angles in a triangle is \(180^{\circ}\). So \(C = 180^{\circ}-A - B\).
Substitute \(A = 49.6^{\circ}\) and \(B=38.9^{\circ}\):
\(C=180^{\circ}-49.6^{\circ}-38.9^{\circ}=91.5^{\circ}\)

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

By the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
So \(a=\frac{c\sin A}{\sin C}\).
Substitute \(c = 26.9\), \(A = 49.6^{\circ}\), \(C = 91.5^{\circ}\):
\(a=\frac{26.9\times\sin(49.6^{\circ})}{\sin(91.5^{\circ})}\)
Since \(\sin(91.5^{\circ})\approx1\) and \(\sin(49.6^{\circ})\approx0.761\), \(a=\frac{26.9\times0.761}{1}\approx20.57\)

Step3: Calculate the area of the triangle

The area formula \(S=\frac{1}{2}ac\sin B\)
Substitute \(a\approx20.57\), \(c = 26.9\), \(B = 38.9^{\circ}\), \(\sin(38.9^{\circ})\approx0.628\)
\(S=\frac{1}{2}\times20.57\times26.9\times0.628\)
\(S=\frac{1}{2}\times20.57\times26.9\times0.628\approx175.5\)

Answer:

\(175.5\)