Find the missing side of the bottom left triangle.

Let's analyze the bottom left triangle. Let's denote the vertices as A, B, and C, and point E lies on side BC. Angle A is 90 degrees, angle B is 30 degrees, angle BEC is 60 degrees, and the length of EC is 7. We need to find the length of BE. First, let's find the angle EAB: $$\angle EAB = 90 - 60 = 30^{\circ}$$ Now, let's find the angle AEB: $$\angle AEB = 180 - 60 = 120^{\circ}$$ In triangle BEC, angle ECB $$= 90 - 60 = 30^{\circ}$$. Now let BE be x. Then $$BE = BC - EC$$ Then using tangent we can write: $$\tan(30) = \frac{AC}{BC}$$ $$\tan(60) = \frac{AC}{EC} = \frac{AC}{7}$$ Therefore $$AC = 7\tan(60)$$ $$AC = 7\sqrt{3}$$ Now we use the other equation: $$\tan(30) = \frac{7\sqrt{3}}{BC}$$ $$BC = \frac{7\sqrt{3}}{\tan(30)} = \frac{7\sqrt{3}}{1/\sqrt{3}} = 7\sqrt{3}\sqrt{3} = 21$$ $$BE = BC - EC = 21 - 7 = 14$$ Answer: 14