Problem:
Equilateral △ABC with side length 14 is rotated about its center by angle θ, where 0<θ≤60∘, to form △DEF. See the figure. The area of hexagon ADBECF is 913. What is tanθ ?
Answer Choices:
A. 53
B. 1153
C. 54
D. 1311
E. 1373
Solution:
Let O be the center of △ABC and △DEF, let P be the foot of the perpendicular from D to AB, and let M be the midpoint of AB.
The condition θ≤60∘ implies that P lies on AM (as opposed to lying on BM ). Furthermore, O is the center of the circle containing points A,B, and D, so by the Inscribed Angle Theorem ∠DOA=2∠DBA. It suffices to compute tan∠DBA and then use the Double Angle Formula to obtain tanθ.
The area of △ABC is 43⋅142=493. Because hexagon ADBECF consists of △ABC plus three copies of △ADB, the area of △ADB is
3913−493=143.
This implies that DP=23. Furthermore, OM=373 and OD=OA=2⋅OM=3143. The Pythagorean Theorem yields
.jpg)