Solution:

To find the area of the triangle, we can drop the altitude to the side of length $12.$

Then we have a right triangle with one leg $12 \div 2 = 6$ and hypotenuse $10.$

The other leg has length $$\sqrt{10^2 - 6^2} = \sqrt{64} = 8.$$

The area of the triangle is then $$\dfrac{8 \cdot 12}{2} = 48.$$

The length of the rectangle is then $48 \div 4 = 12.$ Its perimeter is $$2(4 + 12) = 2 \cdot 16 = 32.$$

Thus, D is the correct answer.