Problem:
Pentagon ABCDE is inscribed in a circle, and ∠BEC=∠CED=30∘. Let AC and BD intersect at point F, and suppose that AB=9 and AD=24. What is BF?
Answer Choices:
A. 1157
B. 1159
C. 1160
D. 1161
E. 1163
Solution:
Note that as ∠BEC and ∠CED are angles inscribed in arcs BC⌢ and CD⌢, so thus ∠BAC=∠CAD=30∘.
This means that AC is an angle bisector of ∠BAD, so we can apply the Angle Bisector Theorem. Also note that
∠BAD=180∘−BCD=180∘−120∘=60∘
By the Law of Cosines,
BD2=AD2+AB2−2⋅AB⋅AD⋅cos∠BAD=92+242−2⋅9⋅24cos60∘=441
so BD=21. Then, by the Angle Bisector Theorem,
BF=AB+ADAB⋅BD=9+249⋅21=1163
leading to an answer of (E) 1163.
The problems on this page are the property of the MAA's American Mathematics Competitions
