<prompt> Acha ABC iwe pembetatu yenye angle BAC 90circ. Mduara umepinda kwa pande AB na AC kwa X na Y mtawalia, hivi kwamba pointi kwenye mduara kinyume cha kipenyo cha X na Y zote ziko upande BC. Kwa kuzingatia kwamba AB 6, pata eneo la sehemu ya duara ambayo iko nje ya pembetatu.asykuagiza olympiad;kuagiza hisabati;kuagiza grafu;ukubwa wa kitengo cm 4;jozi A 0,0;jozi B A kulia;jozi C A juu;jozi O 13, 13;jozi Xprime 13,23;jozi Yprime 23,13;kujazaArcO,13,0,90--Xprime--Yprime--cycle,0.7nyeupe;kuchoraA--B--C--mzunguko;choraMduaraO, 13;chora0,13--23,13;chora13,0--13,23;chora116,0--116,116--0,116;leboA,A, SW;leboB,B, chini;leboC,C, kushoto;leboX,13,0, chini;leboY,0,13, kushoto;asy  <response>: Acha O iwe kitovu cha mduara, na r kipenyo chake, na acha X na Y ziwe pointi zilizo kinyume kipenyo cha X na Y, mtawalia. Tuna OX OY r, na angle XOY 90circ. Kwa kuwa pembetatu XOY na BAC zinafanana, tunaona kwamba AB AC. Acha X iwe chini ya mwinuko kutoka Y hadi overlineAB. Kwa kuwa XBY ni sawa na ABC, na XY r, tuna XB r. Inafuata kwamba AB 3r, hivyo r 2.asykuagiza olympiad;kuagiza hisabati;kuagiza grafu;ukubwa wa kitengo cm 4;jozi A 0,0;jozi B A kulia;jozi C A juu;jozi O 13, 13;jozi Xprime 13,23;jozi Yprime 23,13;kujazaArcO,13,0,90--Xprime--Yprime--cycle,0.7nyeupe;kuchoraA--B--C--mzunguko;choraMduaraO, 13;chora0,13--23,13;chora13,0--13,23;chora23, 0--23, 13;chora116,0--116,116--0,116;leboA,A, SW;leboB,B, chini;leboC,C, kushoto;leboX,13,0, chini;leboY,0,13, kushoto;leboX, 13, 23, NE;leboY, 23, 13, NE;leboX, 23, 0, chini;leboO, O, NE;asyKisha, eneo linalohitajika ni eneo la mduara wa robo ukiondoa lile la pembetatu XOY. Na jibu ni frac 1 4 pi r2 - frac 1 2 r2 pi - 2. 