Answer to Riddle #85: Garden Path

85. A man has a rectangular garden, 55m ('B' in the diagram,) by 40m('A',) and he makes a diagonal path, 1m wide, exactly in the manner indicated in the diagram. What is the area of the path?
Diagram of a garden with a path diagonally across the middle

At first this puzzle looks trivial, just simple geometry. It's not. It's tricky:

Before reading the answer can I interest you in a clue?

Lets label up a diagram so we know what we're talking about. Note the diagonal path, in the corners, does not meet in the way you might expect:

fully labeled diagram of the garden path

We are going to use two relationships to solve for the area of the path: Pythagerous on the triangle (A,B-z,L+y), and that the area of the path + the area of the grass is equal to the total area (A•B).

The area of the path is given by (L+y)•x, if that's not clear then it should be from the following diagram, where we imagine a triangle removed from the bottom left to the top right, showing that it forms a rectangle.

conceptual diagram moving the triangle in the bottom left to the top right so as to demonstrate why the area of the path is given by x(L+y)

So the equation for the areas is as follows:
first equation equating the areas
And the equation using Pythagerous on the △(A,B-z,L+y): second equation performing pythag on triangle formed by (A,B-z,L+y)
From here we can rearrange the first equation in terms of (L+y) and substitute it in to the second. We will ultimate form a quadratic equation for z which we will solve in the usual way, then solve solve for the area of the path by plugging 'z' into the equation where where we equated the areas: first equation rearranged in terms of L+y
second equation with the rearranged first equation substituted in to it.
At this point it's honestly easier to just plug the numbers in, (it's particularly helpful that x=1,) and you should be able rearrange to a quadratic that looks like the following:
final quadratic to be solved for z.
Which solves to give z=1 2⁄3, which we can use in our very first equation. The first equation basically says the area of the path = total area (2200m2,) - the area of the grass. Which using our value of z=1 2⁄3 gives us and area for the path of 66 2⁄3m2

Puzzle comes from a book, 536 Puzzles by Martin Gardner

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