Final answer:
To find DE, we can use the fact that the triangles GFC and AHD are congruent to the shaded triangles HKD and KFD. Therefore, DE = (x*r)/R.
Step-by-step explanation:
To find DE, we can use the fact that the triangles GFC and AHD are congruent to the shaded triangles HKD and KFD. Since AB = 3x and AC = 3R, we can set up a proportion: AB/AC = DE/FC. Substituting the given values, we have 3x/3R = DE/FC. Since FC is tangent to the circle at point B, it is perpendicular to AB. Therefore, FC is the radius of the circle, which we'll denote as r. We can substitute r for FC in the proportion and solve for DE: (3x/3R) = DE/r. Cross-multiplying, we get 3x*r = 3R*DE. Simplifying, we have x*r = R*DE. Dividing both sides of the equation by R, we get (x*r)/R = DE. Therefore, DE = (x*r)/R.