The length of the perpendicular drawn on the smallest side of a scalene triangle does not have any relation with the lengths of the other sides of the triangle. Therefore, the correct answer is option 'C' - No relation.

Explanation:
A scalene triangle is a triangle in which all three sides have different lengths. Let's consider a scalene triangle ABC, where AB is the smallest side. Now, let's draw a perpendicular from vertex A to side BC and denote the point of intersection as D.

The length of the perpendicular AD can vary depending on the angles and lengths of the sides of the triangle. Here are a few scenarios to understand this concept:

1. Triangle with acute angles:
If triangle ABC has all three angles acute, the length of the perpendicular AD will be the shortest distance from point A to side BC. In this case, AD will be the smallest compared to the lengths of the other sides of the triangle.

2. Triangle with a right angle:
If triangle ABC has a right angle at vertex A, then the perpendicular AD will be the length of the altitude from A to side BC. In this case, AD will be the longest compared to the lengths of the other sides of the triangle.

3. Triangle with an obtuse angle:
If triangle ABC has an obtuse angle at vertex A, the length of the perpendicular AD will be between the shortest and longest side of the triangle. It will not have a direct relation with the lengths of the other sides.

From these scenarios, it is clear that the length of the perpendicular drawn on the smallest side of a scalene triangle does not have a fixed relationship with the lengths of the other sides. It can be the smallest, largest, or somewhere in between. Therefore, the correct answer is option 'C' - No relation.