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. π218
C. π29(3−1)
D. π29(3+1)
E. π29(3+3)
Solution:
Let R be the radius of the circle. Then the circumference of the circle is 2πR=3+4+5 so R=π6. The central angle subtended by the arc of length 5 measures 65/π=65π radians. Likewise, the angles subtended by the arcs of length 4 and 3 have measures 32π and 2π 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
