Question

b = 34°42 a = 38.7 b = 31.2 answer boxes within the choice. a. there is only one possible solution for the triangle. the measurements for the remaining angles a and c a a = ° c = ° c = (simplify your answer. round to the nearest (rol degree as needed. round to the nearest as n minute as needed.) b. there are two possible solutions for the triangle. the measurements for the solution with the longer side a₁ = ° c₁ = ° c₁ = (simplify your answer. round to the nearest (rol degree as needed. round to the nearest as n minute as needed.) the measurements for the solution with the shorter sid a₂ = ° c₂ = ° c₂ (simplify your answer. round to the nearest (re degree as needed. round to the nearest a minute as needed.) c. there are no possible solutions for this triangle.

Explanation:

Step1: Convert angle \( B \) to decimal degrees

\( 42'=\frac{42}{60}=0.7^{\circ}\), so \( B = 34.7^{\circ}\)

Step2: Use the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\)

\(\sin A=\frac{a\sin B}{b}\)
Substitute \(a = 38.7\), \(b = 31.2\), \(B=34.7^{\circ}\)
\(\sin A=\frac{38.7\times\sin(34.7^{\circ})}{31.2}\)
\(\sin(34.7^{\circ})\approx0.57\)
\(\sin A=\frac{38.7\times0.57}{31.2}\approx\frac{22.06}{31.2}\approx0.707\)
\(A=\sin^{- 1}(0.707)\approx45^{\circ}\) or \(A = 180^{\circ}-45^{\circ}=135^{\circ}\)
But if \(A = 135^{\circ}\), \(A + B=135^{\circ}+34.7^{\circ}=169.7^{\circ}<180^{\circ}\)

Step3: Calculate angle \(C\) for \(A = 45^{\circ}\)

\(C=180^{\circ}-(A + B)=180^{\circ}-(45^{\circ}+34^{\circ}42')=100^{\circ}18'\)
For \(A = 135^{\circ}\), \(C=180^{\circ}-(135^{\circ}+34^{\circ}42')=10^{\circ}18'\)

Answer:

B. There are two possible solutions for the triangle.
The measurements for the solution with the longer side \(c\):
\(A_{1}=45^{\circ}0'\), \(C_{1}=100^{\circ}18'\)
The measurements for the solution with the shorter side \(c\):
\(A_{2}=135^{\circ}0'\), \(C_{2}=10^{\circ}18'\)