A = 1/2 a *P where a is apothem and P is perimeter.

1/2 * 1/2 y * 10x = 5xy/2: Answer 1.

You can go further by cutting the pentagon into 5 triangles, and cutting each of those into two right triangles with height 1/2 y and base x. The triangles thus formed are 36 (opposite the side length x), 54 (opposite the side length 1/2 y) 90 triangles.

Then tan(36) = x/(1/2y) so x = 1/2 y * tan(36)

Substitute that back into the 5xy/2 gives 5*(1/2 y *tan(36))*y /2 = Answer 2.

tan(54) = y/2x and 2x*tan(54) = y

Then 5xy/2 = 5x*2x*tan(54)/2 = Answer 3.

Those three answers are equivalent.

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