We can use the trigonometric ratios of sine, cosine, and tangent to solve for the missing sides.
Given a = 6 and angle A = 30 degrees, we can use the sine ratio to find b:
sin A = opposite/hypotenuse
sin 30 = b/c
Substituting b = c (since it is a right triangle), we get:
sin 30 = b/b
b = sin 30 * c
We also know that:
a^2 + b^2 = c^2
Substituting the given values, we get:
6^2 + (sin 30 * c)^2 = c^2
Simplifying, we get:
36 + sin^2 30 * c^2 = c^2
36 + (1/4) * c^2 = c^2
3/4 * c^2 = 36
c^2 = 48
c = sqrt(48) = 4sqrt(3)
Substituting c = 4sqrt(3), we get:
b = sin 30 * c = (1/2) * 4sqrt(3) = 2sqrt(3)
Therefore, the missing side lengths are b = 2sqrt(3) and c = 4sqrt(3).