Question

which one of the following sets of data does not determine a unique triangle? choose the correct answer below. a. a = 50°, b = 21, a = 19 b. a = 30°, b = 8, a = 4 c. a = 45°, b = 10, a = 12 d. a = 130°, b = 4, a = 7

Explanation:

Step1: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}\), so \(\sin B=\frac{b\sin A}{a}\)

Step2: Analyze Option A

Given \(A = 50^{\circ}\), \(b = 21\), \(a=19\)
\(\sin B=\frac{21\sin50^{\circ}}{19}\approx\frac{21\times0.7660}{19}\approx0.842\)
Since \(0 < B<180^{\circ}\), \(B\approx57.4^{\circ}\) or \(B\approx122.6^{\circ}\)
Two possible triangles.

Step3: Analyze Option B

Given \(A = 30^{\circ}\), \(b = 8\), \(a = 4\)
\(\sin B=\frac{8\sin30^{\circ}}{4}=\frac{8\times0.5}{4}=1\)
\(B = 90^{\circ}\), one - triangle.

Step4: Analyze Option C

Given \(A=45^{\circ}\), \(b = 10\), \(a = 12\)
\(\sin B=\frac{10\sin45^{\circ}}{12}=\frac{10\times\frac{\sqrt{2}}{2}}{12}\approx\frac{7.07}{12}\approx0.589\)
Since \(a>b\), \(B\) is acute (\(B\approx36^{\circ}\)), one - triangle.

Step5: Analyze Option D

Given \(A = 130^{\circ}\), \(b = 4\), \(a = 7\)
Since \(A>90^{\circ}\) and \(a>b\), one - triangle.

Answer:

A. \(A = 50^{\circ},b = 21,a = 19\)