Problem:
Equilateral β³ABC with side length 14 is rotated about its center by angle ΞΈ, where 0<ΞΈβ€60β, to form β³DEF. See the figure. The area of hexagon ADBECF is 913β. What is tanΞΈ ?
Answer Choices:
A. 53β
B. 1153ββ
C. 54β
D. 1311β
E. 1373ββ
Solution:
Let O be the center of β³ABC and β³DEF, let P be the foot of the perpendicular from D to AB, and let M be the midpoint of AB.
The condition ΞΈβ€60β 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 β DOA=2β DBA. It suffices to compute tanβ DBA and then use the Double Angle Formula to obtain tanΞΈ.
The area of β³ABC is 43βββ 142=493β. Because hexagon ADBECF consists of β³ABC plus three copies of β³ADB, the area of β³ADB is
3913ββ493ββ=143β.
This implies that DP=23β. Furthermore, OM=373ββ and OD=OA=2β OM=3143ββ. The Pythagorean Theorem yields
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