Short Answer 
The summary explains the steps to establish the relationship between radius (r), height (h), and length (L) in a geometric context using key equations. By applying the Pythagorean theorem, it derives the ratio L/r = 8/5, after simplifying the formulated equation.
Step 1: Understand the Relationship Between Variables
In our geometric construction, we have the relationship between radius r, height h, and length L. The key equations to focus on are:
- r + h = L
- h = L – r
These equations help establish how the triangle forms between the circle and square, giving a basis for applying Pythagoras’s theorem.
Step 2: Apply Pythagoras’ Theorem
We know that the triangle involving r, h, and half of L is a right triangle. By using Pythagorean theorem, we have:
- r² = h² + (L/2)²
Next, substitute h with its expression in terms of L and r to formulate the equation:
- r² = (L – r)² + (L/2)²
Step 3: Simplify and Solve for the Ratio
After expanding and simplifying the equation derived from Pythagoras’ theorem, you can arrive at an equation in terms of L and r:
- 8r = 5L
From this, you can easily solve for the ratio L/r to get:
- L/r = 8/5