Answer:
The length of [tex]b[/tex] is approximately 11.7.
Step-by-step explanation:
The sum of internal angles in triangles equals 180°, as we know the measures of angles B and C, we determine the measure of angle A by algebraic means:
[tex]A = 180^{\circ}-B-C[/tex] (1)
([tex]B = 22^{\circ}[/tex], [tex]C = 52^{\circ}[/tex])
[tex]A = 180^{\circ}-22^{\circ}-52^{\circ}[/tex]
[tex]A = 106^{\circ}[/tex]
The length of [tex]b[/tex] is found by the Law of Sine:
[tex]\frac{b}{\sin B} = \frac{a}{\sin A}[/tex]
[tex]b = a\cdot \left(\frac{\sin B}{\sin A} \right)[/tex]
([tex]a = 30[/tex], [tex]A = 106^{\circ}[/tex], [tex]B = 22^{\circ}[/tex])
[tex]b = 30\cdot \left(\frac{\sin 22^{\circ}}{\sin 106^{\circ}} \right)[/tex]
[tex]b \approx 11.7[/tex]
The length of [tex]b[/tex] is approximately 11.7.
