Geometry
# Solving Triangles

Triangle \(ABC\) has angles \(\angle A = 60^{\circ}\) and \( \angle B = 45^{\circ} \), and a side length \(a = 21 \sqrt{6}\). What is the side length \(b\)?

**Details and assumptions**

\(a\) and \(b\) are the lengths of the sides opposite to the vertices \(A\) and \(B\), respectively.

In triangle \(ABC\) above, we are given the side lengths as follows:\[\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c.\] If \(\sin A : \sin B = 5 : 4\) and \(\ a : c = 2 : 5,\) what is \(a : b : c?\)

**Note:** The above diagram is not drawn to scale.

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