We will follow the instructions and then we need first to find the area of any regular n-gon.
I attached one graph so that it is easier to understand.
The n-gon can be divided in n similar isosceles triangles.
So we can find the area of one of these triangles and then multiply by n to get the total area, right?
Let's focus on the triangle OAB then. OA = OB = r, right?
The area of this triangle is the (altitude * AB ) / 2
The angle AOB is by construction of the regular n-gon.
So half this angle is and we can use cosine rule to come up with the altitude:
Using the sine rule we can write that
So, the area of the triangle is
Ok, but wait, we know how to simplify. We can use that
So it gives that the area of one triangle is:
Last step, we need to multiply by n.
Now, let's use a result from Calculus:
How to use it here? Just notice that
Hope this helps.
Do not hesitate if you need further explanation.
Please refer to the attached diagram.
Point A can be assigned x-coordinate "p". Then its y-coordinate is 6p^2. The slope at that point is y'(p) = 12p.
Point B can be assigned x-coordinate "r". Then its y-coordinate is 6r^2. The slope at that point is y'(r) = 12r.
We want the slopes at those points to have a product of -1 (so the tangents are perpendicular). This means ...
(12p)(12r) = -1
r = -1/(144p)
The slope of line AB in the diagram is the ratio of the differences of y- and x-coordinates:
slope AB = (ry -py)/(rx -px) = (6r^2 -6p^2)/(r -p) = 6(r+p) . . . . simplified
The slope of AB is also the tangent of the sum of these angles: the angle AC makes with the x-axis and angle CAB. The tangent of a sum of angles is given by ...
tan(α+β) = (tan(α) +tan(β))/1 -tan(α)·tan(β))
Of course the slope of a line is equal to the tangent of the angle it makes with the x-axis. The tangent of angle CAB is 2 (because the aspect ratio of the rectangle is 2). This means we can write ...
slope AB = ((slope AC) +2)/(1 -(slope AC)(2))
So, now we can figure the coordinates of points A and B, and the distance between them. That distance is given by the Pythagorean theorem as ...
d^2 = (6r^2 -6p^2)^2 +(r -p)^2
d^2 = (6(1/6)^2 -6(-1/24)^2)^2 +(1/6 +1/24)^2 = 25/1024 +25/576 = 625/9216
Because of the aspect ratio of the rectangle, the area is 2/5 of this value, so we have ...
Rectangle Area = (2/5)(625/9216) = 125/4608 = a/b
Then a+b = 125 +4608 = 4733.
Comment on the solution
The point of intersection of the tangent lines is a fairly messy expression, and that propagates through any distance formulas used to find rectangle side lengths. This seemed much cleaner, though maybe not so obvious at first.