Dynamic Geometry: P118

From the compiled calculations in Dynamic Geometry: P96 Series , we know that the length of the yellow dividing chord is $12$ , and the largest rectangle inscribed in a triangle is one with half the width and height of the triangle.

Let the height of the upper and lower triangles be $h$ and $h'$ respectively. Then the area of the upper triangle is $A_\triangle = \frac 12 \times 12 h = 6 h$ ; the area of the rectangle it inscribed $A_\square = \frac h2 \times \frac {12}2 = 3 h$ ; therefore the purple area $A=A_\triangle - A_\square = 6h - 3h = 3h$ . Similarly, the cyan area in the lower triangle $A' = 3h'$ . Then the ratio of the areas:

$\frac A{A'} = \frac {3h}{3h'} = \frac h{h'}$

It is given that the yellow chord divide the circle into two circular segments of heights $4$ and $9$ . When $\dfrac h{h'} = \frac 49$ , the centroids of the two triangles and the centers of the two rectangles are colinear with the center of the circle. Then $h=4$ and $h'=9$ , and the centers of the upper and lower rectangles are $\dfrac h4 = 1$ and $\dfrac {h'}4 = \dfrac 94$ respectively. The distance between the two centers of the rectangles is $1 + \dfrac 94 = \dfrac {13}4$ . Therefore $\sqrt{p-q}-1 = \sqrt{13-4}-1 = \boxed 2$ .