• Realize that one can draw a vertical line from the top to the tangent point, due to it being a quarter circle.
• Set the origin at the tangent point
• Use point (5,1) = G being part of the circle to obtain the radius of the circle R by using

x^2 + (y-R)^2 = R^2

• With R = 13, we can use Pythagoras to find that the length of AB = 12.
• We can find the angle t using the 5-12-13 triangle denoted by ABC corners.
• Use the \cos(t) = 12/13 to find that \sin(90-t)=12/13=a/12, hence a=12.
• Apply Pythagoras on triangle ADE to find that 13-b=5 and thus b=8.
• Use the ratios between sides of similar triangles ABC and FHG to find that the length of FG equals 91/12.
• You have all the information to find the area of the big rectangle (136) and the remaining triangle forming the rest of the shaded region (\sim 35.21).
• Add up the two values to find the final answer.