The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of pi.
Let the midpoint of chord AB = D
Let the center of the larger circle =O
Connect AO and BO to form triangle ABO
Draw a perpendicular line from O to D
Now, you have 2 congruent right triangles with:
Angle OAB = OBA = 45 degrees
Cos(45) = 40/BA(Radius)
BA =56.568 - Radius of the large circle
Area =(56.568)^2 x Pi =~3200 pi
Tan(45) =DO(radius of the small circle) / 40
Radius = 40
Area of the smaller circle =40^2 x pi =1600 pi
3200 pi - 1600 pi =1600 pi - area of the shaded region.