Question 1149586
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Each of the line segments parallel to one side of the triangle, along with portions of the other two sides, forms a triangle that is similar to the whole triangle.<br>
The triangle formed by the longest of those parallel line segments has an area 5/6 of the area of the original.  That is, the ratio of the areas is 5:6.<br>
In any two similar figures, if the ratio of the areas is A:B, then the ratio of similarity is sqrt(A):sqrt(B).  So in this problem the ratio of similarity is sqrt(5):sqrt(6).<br>
And so the length of the longest of the line segments parallel to the side with length 24 is<br>
{{{24*(sqrt(5)/sqrt(6))}}}<br>
Convert that to another form if required.<br>