Question

2 multiple choice 1 point the given measurements may or may not determine a triangle. if not, then state that no triangle is formed. if a triangle is formed, then use the law of sines to solve the triangle, if it is possible, or state that the law of sines cannot be used. a = 52°, a = 7, b = 65° c = 63°, b ≈ 8.1, c ≈ 7.9 no triangle is formed. c = 63°, b ≈ 8.1, c ≈ 6.9 c = 27°, b ≈ 6.1, c ≈ 6.9

Explanation:

Step1: Find angle \(C\)

In a triangle, \(A + B + C=180^{\circ}\). Given \(A = 52^{\circ}\) and \(B = 65^{\circ}\), then \(C=180^{\circ}-(A + B)=180^{\circ}-(52^{\circ}+ 65^{\circ})=63^{\circ}\).

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

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 7\), \(A = 52^{\circ}\), \(B = 65^{\circ}\). From \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we have \(b=\frac{a\sin B}{\sin A}\).
\(\sin A=\sin52^{\circ}\approx0.788\), \(\sin B=\sin65^{\circ}\approx0.906\). Then \(b=\frac{7\times0.906}{0.788}=\frac{6.342}{0.788}\approx8.1\).

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

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), \(\sin C=\sin63^{\circ}\approx0.891\). Then \(c=\frac{a\sin C}{\sin A}=\frac{7\times0.891}{0.788}=\frac{6.237}{0.788}\approx7.9\).

Answer:

C = 63°, b ≈ 8.1, c ≈ 7.9 (First option)