The ratio of lengths of two rods A and B is given as 1:2, and the ratio of their radii is given as 2:1. We are required to find the ratio of rigidity of rods A and B.

The rigidity of a rod is determined by its Young's modulus (E), which is a measure of its stiffness or resistance to deformation. Young's modulus is given by the formula:

E = (F/A) / (ΔL/L)

where F is the force applied, A is the cross-sectional area of the rod, ΔL is the change in length, and L is the original length of the rod.

To find the ratio of rigidity of rods A and B, we need to compare their Young's moduli.

Let's assume the length of rod A is L and the radius is r. Therefore, the length of rod B will be 2L and the radius will be 2r, according to the given ratios.

1. Calculate the cross-sectional areas of rods A and B:
- The cross-sectional area of rod A (A_A) is πr^2.
- The cross-sectional area of rod B (A_B) is π(2r)^2 = 4πr^2.

2. Calculate the change in length of rods A and B:
- Let's assume a force F is applied to both rods, resulting in a change in length ΔL.
- The change in length of rod A (ΔL_A) will be ΔL.
- The change in length of rod B (ΔL_B) will also be ΔL, as the force and material are the same.

3. Calculate the Young's modulus of rods A and B:
- Young's modulus of rod A (E_A) = (F/A_A) / (ΔL/L) = (F/πr^2) / (ΔL/L)
- Young's modulus of rod B (E_B) = (F/A_B) / (ΔL/2L) = (F/4πr^2) / (ΔL/2L)

4. Simplify the ratios:
- E_A / E_B = [(F/πr^2) / (ΔL/L)] / [(F/4πr^2) / (ΔL/2L)]
= (F/πr^2) / (F/4πr^2) * (ΔL/2L) / (ΔL/L)
= 4 * (ΔL/2L)
= 2 * (ΔL/L)

From the above derivation, we can conclude that the ratio of rigidity of rods A and B is 2:1, which is the same as the ratio of their radii.