Question

given the measures ( a = 10 ), ( b = 40 ), and ( a = 30^{circ} ), how many triangles can possibly be formed?
done
given the measures ( b = 10 ), ( c = 8.9 ), and ( b = 63^{circ} ), how many triangles can possibly be formed?
done
no triangle
( a lt h )
one right triangle
( a = h )
one triangle
( a gt h ), ( a gt b )
two triangles
( a gt h ), ( a lt b )

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{\sin B}{b}=\frac{\sin C}{c}\). Substitute \(b = 10\), \(c=8.9\), and \(B = 63^{\circ}\) into the formula: \(\sin C=\frac{c\sin B}{b}\).

Step2: Calculate \(\sin C\)

\(\sin C=\frac{8.9\times\sin63^{\circ}}{10}\). Since \(\sin63^{\circ}\approx0.891\), then \(\sin C=\frac{8.9\times0.891}{10}=\frac{7.9299}{10} = 0.79299\).

Step3: Analyze the value of \(C\)

Since \(0<\sin C = 0.79299<1\), and \(c < b\) (\(8.9<10\)), there is one triangle.

Answer:

One triangle