Question

two sides and an angle are given. determine whether the given information results in one triangle, two triangles, or no triangle at all. solve any resulting triangle(s)
a = 10, b = 9, a = 20°
select the correct choice and, if necessary, fill in the answer boxes to complete your choice
a. a single triangle is produced, where b ≈ □°, c ≈ □°, and c ≈ □
(type integers or decimals rounded to two decimal places as needed.)

Explanation:

Step1: Use the Law of Sines to find angle \( B \)

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substituting the given values \(a = 10\), \(b = 9\), and \(A=20^{\circ}\), we get \(\frac{10}{\sin20^{\circ}}=\frac{9}{\sin B}\).
Solving for \(\sin B\), we have \(\sin B=\frac{9\sin20^{\circ}}{10}\).
Calculating \(9\sin20^{\circ}\approx9\times0.3420 = 3.078\), then \(\sin B=\frac{3.078}{10}=0.3078\).
So \(B=\sin^{-1}(0.3078)\approx18.00^{\circ}\) (since \(0^{\circ}b\)).

Step2: Find angle \( C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-A - B\).
Substituting \(A = 20^{\circ}\) and \(B\approx18.00^{\circ}\), we get \(C=180^{\circ}-20^{\circ}-18^{\circ}=142.00^{\circ}\).

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

Using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we have \(c=\frac{a\sin C}{\sin A}\).
Substituting \(a = 10\), \(A = 20^{\circ}\), and \(C = 142^{\circ}\), \(\sin C=\sin142^{\circ}\approx0.6157\), \(\sin A=\sin20^{\circ}\approx0.3420\).
Then \(c=\frac{10\times0.6157}{0.3420}\approx17.99\).

Answer:

A. A single triangle is produced, where \(B\approx18.00^{\circ}\), \(C\approx142.00^{\circ}\), and \(c\approx17.99\)