Problem:
A circle with center O has area 156Ο. Triangle ABC is equilateral, BC is a chord on the circle, OA=43β, and point O is outside β³ABC. What is the side length of β³ABC?
Answer Choices:
A. 23β
B. 6
C. 43β
D. 12
E. 18
Solution:
The radius of circle O is 156β>43β=OA, so A is inside the circle. Let s be the side length of β³ABC, let D be the foot of the altitude from A, and let OE be the radius through A. This radius is perpendicular to BC and contains D, so OD=OB2βBD2β=156β41βs2β. If A is on DE, then β BAC>β BEC>90β, an impossibility. Therefore A lies on OD, and OA=ODβAD, that is,
43β=156β41βs2ββ23ββs
Rearranging terms and squaring both sides leads to the quadratic equation s2+12sβ108=0, and the positive solution is s=(B)6β.
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