Henry Ernest Dudeney/Puzzles and Curious Problems/231 - The Rose Garden/Solution

by : $231$

 * The Rose Garden
 * A man has a rectangular garden, and wants to make exactly half of it into a large bed of roses,
 * with a gravel path of uniform width round it.
 * Can you find a general rule that will apply equally to any rectangular garden, whatever its proportions?


 * All the measurements must be made in the garden.
 * A plain ribbon, no shorter than the length of the garden, is all the material required.

Solution

 * Dudeney-Puzzles-and-Curious-Problems-231-solution.png

Construct $AD$ one quarter the length of $AB$.

Construct $AF$ and $DE$ one quarter the length of $BC$.

Construct $EG = DF$.

Then $AG$ is the required width of the path.

Proof
Let the length and breadth of the garden be $a$ and $b$.

Let $c$ be the width of the path.

We have:

It is apparent that it is the negative square root of $\sqrt {a^2 + b^2}$ we need in the above, as the positive one would result in the path being wider than the garden.

Hence:


 * $c = \dfrac {a + b - \sqrt {a^2 + b^2} } 4$

Now we investigate the geometry.

Let $a = AB$ and $b = BC$.

We have that:

and it is seen that $AG$ is the same as what was calculated algebraically above.