Question

how many triangles exist that fit the following criteria?

b = 130°, b = 7, a = 6

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). We know \(B = 130^{\circ}\), \(b = 7\), and \(a=6\). Substitute these values into the formula: \(\sin A=\frac{a\sin B}{b}\).

$$ \sin A=\frac{6\sin130^{\circ}}{7} $$

Since \(\sin130^{\circ}=\sin(180 - 50)^{\circ}=\sin50^{\circ}\approx0.766\), then \(\sin A=\frac{6\times0.766}{7}\approx\frac{4.596}{7}\approx0.657\).

Step2: Analyze the value of \(A\)

Since \(a < b\) and \(B = 130^{\circ}\) (an obtuse angle). When we find \(\sin A\approx0.657\), \(A=\sin^{- 1}(0.657)\approx41.1^{\circ}\). Also, the supplement of \(A\), \(A'=180^{\circ}-41.1^{\circ} = 138.9^{\circ}\). But \(B + A'=130^{\circ}+138.9^{\circ}=268.9^{\circ}>180^{\circ}\).

Answer:

1