The perimeter of triangle abc is 82 inches. the triangle is rotated about a line that passes through points b and d.

Given an isosceles triangle with a leg length of 24 inches and a perimeter of 82 inches.This means that the length of the other side is 82 - 2(24) = 82 - 48 = 34 inches
Given that the triangle is rotated about a line that passes through points B and D, point D being the midpoint of line AC.
Then, the resulting three-dimensional figure will be a cone with a base radius equal to the length of line AD (i.e. half of the length of line AC being the other side of the isosceles triangle).
The base radius of the resulting cone is 1/2 (34) = 17 inches.
Therefore, the resulting three-dimensional figure is a cone with base radius of 17 inches.