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 to find angle \( B \)

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substitute \(A = 43.5^{\circ}\), \(a = 8.9\)m, and \(b = 10.2\)m into the formula:
\(\sin B=\frac{b\sin A}{a}=\frac{10.2\times\sin43.5^{\circ}}{8.9}\)
\(\sin43.5^{\circ}\approx0.688\), so \(\sin B=\frac{10.2\times0.688}{8.9}\approx\frac{7.018}{8.9}\approx0.789\)
\(B=\sin^{- 1}(0.789)\approx52.1^{\circ}\) or \(B = 180^{\circ}-52.1^{\circ}=127.9^{\circ}\)

Step2: Check for the two - case scenario

Case 1: When \(B_1 = 52.1^{\circ}\)
\(C_1=180^{\circ}-A - B_1=180^{\circ}-43.5^{\circ}-52.1^{\circ}=84.4^{\circ}\)
Using the Law of Sines \(\frac{c_1}{\sin C_1}=\frac{a}{\sin A}\), \(c_1=\frac{a\sin C_1}{\sin A}\)
\(\sin84.4^{\circ}\approx0.995\), \(c_1=\frac{8.9\times0.995}{0.688}\approx\frac{8.856}{0.688}\approx12.9\)m

Case 2: When \(B_2 = 127.9^{\circ}\)
\(C_2=180^{\circ}-A - B_2=180^{\circ}-43.5^{\circ}-127.9^{\circ}=8.6^{\circ}\)
Using the Law of Sines \(\frac{c_2}{\sin C_2}=\frac{a}{\sin A}\), \(\sin8.6^{\circ}\approx0.150\)
\(c_2=\frac{8.9\times0.150}{0.688}\approx\frac{1.335}{0.688}\approx1.9\)m

Answer:

B. There are two possible solutions for the triangle
The measurements for the solution with the longer side \(c\) are as follows:
\(B_1 = 52.1^{\circ}\) (Round to the nearest tenth as needed)
\(C_1 = 84.4^{\circ}\) (Round to the nearest tenth as needed)
\(c_1 = 12.9\)m (Round to the nearest tenth as needed)
The measurements for the solution with the shorter side \(c\) are as follows:
\(B_2 = 127.9^{\circ}\) (Round to the nearest tenth as needed)
\(C_2 = 8.6^{\circ}\) (Round to the nearest tenth as needed)
\(c_2 = 1.9\)m (Round to the nearest tenth as needed)