Question

state whether the given measurements determine zero, one, or two triangles.
c = 30°, a = 20, c = 10
two
zero
one

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, \(\frac{\sin A}{a}=\frac{\sin C}{c}\). Substitute \(a = 20\), \(c = 10\), and \(C=30^{\circ}\). So \(\frac{\sin A}{20}=\frac{\sin30^{\circ}}{10}\).

Step2: Solve for \(\sin A\)

Since \(\sin30^{\circ}=\frac{1}{2}\), then \(\sin A=\frac{20\times\sin30^{\circ}}{10}\). Substitute \(\sin30^{\circ}=\frac{1}{2}\), we get \(\sin A=\frac{20\times\frac{1}{2}}{10}= 1\).

Step3: Analyze the value of \(A\)

Since \(\sin A = 1\), then \(A = 90^{\circ}\) (because \(A\in(0^{\circ},180^{\circ})\) and \(\sin90^{\circ}=1\)).

Answer:

One