Quick Solution

This problem can be solved by setting up an algebraic equation based on the Pythagorean theorem and the information about the "distance saved".

1. Set Up the Equations

• Let the longer side of the rectangle be l and the shorter side be s.
• The distance when walking along the two adjacent sides is (l + s).
• The shortcut distance along the diagonal is d. From the Pythagorean theorem, we know that d = √(l² + s²).
• The "distance saved" is the difference between these two paths: (l + s) - d.

The problem states that this saved distance is equal to half the longer side. This gives us our main equation:

(l + s) - √(l² + s²) = l / 2

2. Solve for the Ratio (s/l)

- First, we rearrange the equation to isolate the square root term on one side:
l + s - l/2 = √(l² + s²)
l/2 + s = √(l² + s²)

- Next, we square both sides of the equation to eliminate the square root:
(l/2 + s)² = l² + s²
(l²/4) + (2 × l/2 × s) + s² = l² + s²
l²/4 + ls + s² = l² + s²

- The s² terms cancel out on both sides, leaving:
l²/4 + ls = l²

- Now, we solve for the relationship between l and s:
ls = l² - l²/4
ls = (3/4)l²

- Finally, to find the required ratio, we divide both sides by l²:
s/l = 3/4

Final Answer: The ratio of the shorter side of the rectangle to the longer side is 3/4.