Final answer:
The length of the longest of the line segments dividing the triangle's area into six equal parts, parallel to the base which is 24 cm, is also 24 cm because the areas' division is in equal parts, and the lengths maintain the mean proportional sequence.
Step-by-step explanation:
To find the length of the longest of the line segments that divides the area of a triangle into six equal parts with the base being 24 cm, we can use the properties of similar triangles.
When line segments are drawn parallel to the base of a triangle dividing it into equal areas, the lengths of these segments form a sequence of numbers in which each is the mean proportional between the base of the triangle and the segment preceding it. Specifically, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Since the area is divided into 6 equal parts, the ratio of the areas of the triangles formed by the segments is 1:1, and thus the square of the ratio of their corresponding sides will also be 1:1. Therefore, the length of the longest segment, which is the one closest to the base, will be the same as the base itself, which is 24 cm.