Question

find the missing parts of the triangle.

a. ( a = 45^{circ}, c = 75^{circ}, c = 3sqrt{3}+3 )
b. ( a = 45^{circ}, c = 75^{circ}, c = 3sqrt{3} )
c. ( a = 75^{circ}, c = 45^{circ}, c = 3sqrt{3}+3 )
d. no such triangle exists.

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, let \(a = 3\sqrt{6}\), \(b = 6\), and \(B=60^{\circ}\). Then \(\frac{3\sqrt{6}}{\sin A}=\frac{6}{\sin60^{\circ}}\).

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Since \(0^{\circ}\lt A\lt180^{\circ}\), \(A = 45^{\circ}\) or \(A = 135^{\circ}\). But if \(A = 135^{\circ}\), then \(A + B=135^{\circ}+ 60^{\circ}=195^{\circ}\gt180^{\circ}\), so \(A = 45^{\circ}\).

Step2: Find angle \(C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C=180^{\circ}-A - B\). Substituting \(A = 45^{\circ}\) and \(B = 60^{\circ}\), we get \(C=180^{\circ}-45^{\circ}-60^{\circ}=75^{\circ}\).

Step3: Find side \(c\)

Using the Law of Sines again, \(\frac{c}{\sin C}=\frac{b}{\sin B}\). So \(c=\frac{b\sin C}{\sin B}\).

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Answer:

A. \(A = 45^{\circ},C = 75^{\circ},c = 3\sqrt{3}+3\)