Solution To Basic Mensa Geometry Problem

 

First redraw the figure as shown above, such that radius DC is tangent to the smaller circle on the side opposite BC.  Here, note that MC is the axis of symmetry for the sector DBC.  Therefore, MC passes through the center O of the smaller circle.

Next draw line segment ON to create the right triangle ONC. By the Pythagorean theorem we can write:

ON ²   +   NC ²  = OC ²

^Or:

7 ²   +   (R - 8) ²  = (R - 7) ²

Here R is the radius of the circle quadrant.  Then:

49 +  R ²   - 16R  + 64  =  R ²  -14R  + 49

And:   - 16R + 64 = - 14R

2R = 64

R = 32 units