Final answer:
The second largest side of a pentagon, which is 25% of the perimeter, corresponds to one of the five consecutive even integer lengths of the pentagon's sides. Setting an algebraic expression for the perimeter and equating it with the given percentage, it is determined that the shortest side is 4 units long.
Step-by-step explanation:
The student's question is related to the properties of polygons and the application of algebraic expressions in geometry. The second largest side of the pentagon is given to be 25% of the whole perimeter, which implies that it represents one of five consecutive even integers that make up the sides of the pentagon. Let's call the shortest side 'x'. Since the sides are consecutive even integers, the sides can be represented as x, x+2, x+4, x+6, and x+8. The second largest side hence is x+6.
The perimeter P of the pentagon is the sum of its sides, so P = x + (x+2) + (x+4) + (x+6) + (x+8). Knowing that the second largest side (x+6) is 25% of P, we have:
0.25P = x + 6
We can now express the perimeter P in terms of x:
P = 5x + 20
Now we solve for x using the 25% relationship:
0.25(5x + 20) = x + 6
1.25x + 5 = x + 6
Subtract x from both sides:
0.25x + 5 = 6
Subtract 5 from both sides:
0.25x = 1
Finally, divide by 0.25:
x = 4
The length of the shortest side is 4 units.