Three straight metal rods have an average (arithmetic mean)

We know that there are 3 metal rods with an average length of 77 inches.
Total length=77*3=231 inches
Shortest rod length=65 inches
the sum of lengths of the two longer rods=231-65=166
Since, the longer rods have to be longer than 65, the smallest value one of these longer values could have is 66.

Now the question asks us for the longest median value. The median value is the middle value when the three rods are arranged in ascending order of lengths.

There are two ways of finding these values
1. Since we need to find the largest middle value, the two longer rods need to be of equal length. If both values are equal in length the middle value will be the largest possible value and the longest rod will be of the same length as the rod of median length. Hence, 2x =166 implies x=83.

Maximum length of the median rod=83

2. Testing the values given
Let us test the values from the bottom

e=89
166-89=77. Since 77 is less than 89, it cannot be the median value. WRONG

d=83
166-83=83. Both the rods are of the same length. Hence, this is the largest possible middle value.