Problem:
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?
Answer Choices:
A. 6
B. Ο218β
C. Ο29β(3ββ1)
D. Ο29β(3β+1)
E. Ο29β(3β+3)
Solution:
Let R be the radius of the circle. Then the circumference of the circle is 2ΟR=3+4+5 so R=Ο6β. The central angle subtended by the arc of length 5 measures 65β/Ο=65Οβ radians. Likewise, the angles subtended by the arcs of length 4 and 3 have measures 32Οβ and 2Οβ radians, respectively. The area of the given triangle is the sum of the areas of the three triangles into which it is partitioned by the radii to its vertices, so the answer is
