Question

solve each triangle abc that exists.
a = 43.5° a = 8.9 m b = 10.2 m
select the correct choice below and, if necessary, fill in the answer boxes within the choice.
○ a. there is only one possible solution for the triangle.
the measurements for the remaining angles b and c and side c are as follows.
b = □° (round to the nearest tenth as needed.)
c = □° (round to the nearest tenth as needed.)
c = □ (round to the nearest tenth as needed.)
○ b. there are two possible solutions for the triangle.
the measurements for the solution with the longer side c are as follows.
b₁ = □° (round to the nearest tenth as needed.)
c₁ = □° (round to the nearest tenth as needed.)
c₁ = □ (round to the nearest tenth as needed.)
the measurements for the solution with the shorter side c are as follows.
b₂ = □° (round to the nearest tenth as needed.)
c₂ = □° (round to the nearest tenth as needed.)
c₂ = □ (round to the nearest tenth as needed.)
○ c. there are no possible solutions for this triangle.

Explanation:

Step1: Use the Law of Sines

By the Law of Sines, \(\frac{\sin B}{b}=\frac{\sin A}{a}\). Substitute \(A = 43.5^{\circ}\), \(a = 8.9\) m, and \(b = 10.2\) m: \(\sin B=\frac{b\sin A}{a}=\frac{10.2\times\sin43.5^{\circ}}{8.9}\).
Calculate \(\sin43.5^{\circ}\approx0.688\), then \(\sin B=\frac{10.2\times0.688}{8.9}\approx0.791\).
So \(B=\sin^{- 1}(0.791)\approx52.3^{\circ}\) or \(B = 180^{\circ}-52.3^{\circ}=127.7^{\circ}\).

Step2: Check for valid triangles

Case 1: If \(B = 52.3^{\circ}\)
\(C=180^{\circ}-A - B=180^{\circ}-43.5^{\circ}-52.3^{\circ}=84.2^{\circ}\)
Again, by the Law of Sines, \(c=\frac{a\sin C}{\sin A}=\frac{8.9\times\sin84.2^{\circ}}{\sin43.5^{\circ}}\). Since \(\sin84.2^{\circ}\approx0.995\), \(c=\frac{8.9\times0.995}{0.688}\approx12.9\) m.
Case 2: If \(B = 127.7^{\circ}\)
\(C=180^{\circ}-A - B=180^{\circ}-43.5^{\circ}-127.7^{\circ}=8.8^{\circ}\)
By the Law of Sines, \(c=\frac{a\sin C}{\sin A}=\frac{8.9\times\sin8.8^{\circ}}{\sin43.5^{\circ}}\). Since \(\sin8.8^{\circ}\approx0.153\), \(c=\frac{8.9\times0.153}{0.688}\approx2.0\) m.

Answer:

B. There are two possible solutions for the triangle.
The measurements for the solution with the longer side \(c\) are as follows.
\(B_{1}=52.3^{\circ}\), \(C_{1}=84.2^{\circ}\), \(c_{1}=12.9\) m
The measurements for the solution with the shorter side \(c\) are as follows.
\(B_{2}=127.7^{\circ}\), \(C_{2}=8.8^{\circ}\), \(c_{2}=2.0\) m