1989 AHSME Problems/Problem 19
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?
Solution
The three arcs make up the entire circle, so the circumference of the circle is and the radius is . Also, the lengths of the arcs are proportional to their corresponding central angles. Thus, we can write the values of the arcs as , , and for some . By Circle Angle Sum, we obtain . Solving yields . Thus, the angles of the triangle are , , and . Using , we obtain . Substituting for and evaluating yields .
-Solution by thecmd999
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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