Question

find the missing parts of the triangle.

oa. b = 60°, c = 90°, b = 4√3
ob. b = 60°, c = 90°, b = 4
oc. no such triangle exists.
od. b = 90°, c = 60°, b = 4√3

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, \(a = 8\), \(A=30^{\circ}\), \(c = 4\). So \(\frac{8}{\sin30^{\circ}}=\frac{4}{\sin C}\).
Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{8}{\frac{1}{2}}=\frac{4}{\sin C}\), which simplifies to \(16=\frac{4}{\sin C}\), and then \(\sin C=\frac{4}{16}=\frac{1}{4}\). But if we assume \(b = 4\sqrt{3}\) (from option A), \(\frac{8}{\sin30^{\circ}}=\frac{4\sqrt{3}}{\sin B}\). \(\frac{8}{\frac{1}{2}} = 16\), so \(\sin B=\frac{4\sqrt{3}}{16}=\frac{\sqrt{3}}{4}
eq\sin60^{\circ}=\frac{\sqrt{3}}{2}\). Wait, another approach: using the triangle - side relationship \(a>c\) ( \(a = 8\), \(c = 4\) ), \(A = 30^{\circ}\). By the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\), \(\sin C=\frac{c\sin A}{a}\). Substituting \(a = 8\), \(c = 4\), \(A = 30^{\circ}\), we get \(\sin C=\frac{4\times\sin30^{\circ}}{8}=\frac{4\times\frac{1}{2}}{8}=\frac{1}{4}\). Also, using the triangle - angle sum \(A + B + C=180^{\circ}\). If we assume the values in option A: \(A = 30^{\circ}\), \(B = 60^{\circ}\), \(C = 90^{\circ}\). Then by the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). \(\frac{a}{\sin30^{\circ}}=\frac{b}{\sin60^{\circ}}=\frac{c}{\sin90^{\circ}}\). If \(a = 8\) (opposite \(A = 30^{\circ}\)), \(\frac{8}{\frac{1}{2}}=16\). If \(b\) (opposite \(B = 60^{\circ}\)) then \(b = 16\times\sin60^{\circ}=16\times\frac{\sqrt{3}}{2}=8\sqrt{3}
eq4\sqrt{3}\). Wait, correct way: \(a = 8\), \(A = 30^{\circ}\), \(c = 4\). By the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\), \(\sin C=\frac{c\sin A}{a}\). Substitute \(a = 8\), \(c = 4\), \(A = 30^{\circ}\), \(\sin C=\frac{4\times\frac{1}{2}}{8}=\frac{1}{4}\). Also, \(a>c\) implies \(A>C\). But if \(C = 90^{\circ}\) (from option A), \(A = 30^{\circ}\), then \(a\) (opposite \(A\)) should be less than \(c\) (opposite \(C\)) since in a triangle, the larger side is opposite the larger angle. But \(a = 8\), \(c = 4\), \(a>c\) and if \(C = 90^{\circ}\), \(A = 30^{\circ}\), this is a contradiction.

Answer:

C. No such triangle exists.