1983 AHSME Problems/Problem 30
Problem
Distinct points and are on a semicircle with diameter and center . The point is on and . If , then equals
Solution
Since , quadrilateral is cyclic (as shown below) by the converse of the theorem "angles inscribed in the same arc are equal".
Since , , so, using the fact that opposite angles in a cyclic quadrilateral sum to , we have . Hence .
Since , triangle is isosceles, with . Now, . Finally, again using the fact that angles inscribed in the same arc are equal, we have .
See Also
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