Zoom International Math League

Question

In the diagram below, tangent AB¯¯¯¯¯¯¯¯AB¯ is perpendicular to diameter CD¯¯¯¯¯¯¯¯CD¯ at CC. EE is the intersection of BD¯¯¯¯¯¯¯¯BD¯ with the circle.

EF¯¯¯¯¯¯¯¯EF¯ is a diameter with AA, EE, OO, and FF all collinear. If the measure of ∠B∠B is 12∘12∘ more than the measure of ∠A∠A, what is the measure of ∠B∠B in degrees?

Solution

Let ∠A∠A have measure xx and ∠B∠B measure yy. Thus y−x=12y−x=12.

EF¯¯¯¯¯¯¯¯EF¯ is a diameter, so arcs ECˆEC^ and CFˆCF^ add up to 180∘180∘. Therefore,

Similarly,Combined with y−x=12y−x=12 we can solve for x=22x=22 and y=34y=34. Thus ∠B∠B measures 34∘34∘.