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To find the area of the triangular garden, we can use the formula for the area of a triangle: A = (1/2) * base * height.
In this case, the shortest side length (x-2y ft.) will be the base of the triangle, and the middle length side (x+6y ft.) will be considered as the height.
So, the area of the garden can be calculated as:
A = (1/2) * (x-2y) * (x+6y)
Now, let's multiply this out and simplify it:
A = (1/2) * (x^2 + 6xy - 2xy - 12y^2)
A = (1/2) * (x^2 + 4xy - 12y^2)
A = (1/2)x^2 + 2xy - 6y^2
Therefore, the area of the garden is (1/2)x^2 + 2xy - 6y^2 square feet.