HMMT 十一月 2010 · 冲刺赛 · 第 17 题

英文解析

17. [ 9 ] A triangle with side lengths 5 , 7 , 8 is inscribed in a circle C . The diameters of C parallel to thesides of lengths 5 and 8 divide C into four sectors. What is the area of either of the two smaller ones?
    Answer: π Let 4 P QR have sides p = 7, q = 5, r = 8. Of the four sectors determined by 49
    the diameters of C that are parallel to P Q and P R , two have angles equal to P and the other two 18
    have angles equal to π − P . We first find P using the law of cosines: 49 = 25 + 64 − 2(5)(8) cos P
    1 π πimplies cos P = implies P = . Thus the two smaller sectors will have angle . Next we find the
    2 3 3
    circumradius of 4 P QR using the formula R = , where [ P QR ] is the area of 4 P QR . By Heron’spqr
    4[ P QR ]
    √
    √
    5 · 7 · 8 7
    √ √
    Formula we have [ P QR ] = 10(5)(3)(2) = 10 3; thus R = = . The area of a smaller sector
    4(10 3) 3
    ( )
    ( )2
    π/ 3
    2 π 7 49
    √
    is thus πR = = π
    2 π 6 18 3