<prompt> Katika mchoro unaokaribiana, miduara miwili yenye radii 8 na 6 imechorwa na vituo vyao vya 12 kando. Kwa P, mojawapo ya pointi za makutano, mstari umechorwa kwa njia ambayo chords QP na PR zina urefu sawa. Tafuta mraba wa urefu wa QP.asyukubwa160; defaultpenlinewidth.8ptfontsize11pt; dotfactor3; jozi O10,0, O212,0; njia C1MduaraO1,8, C2MduaraO2,6; jozi PintersectionpointsC1,C20; njia C3MduaraP,sqrt130; jozi QintersectionpointsC3,C10; jozi Rvipimo vya makutanoC3,C21; kuchoraC1; kuchoraC2; kuchoraO2--O1; nukta O1; nukta O2; choraQ--R; leboQ,Q,NW; leboP,P,1.5dir80; leboR,R,NE; lebo12, waypointO1--O2,0.4,S;asy  <response>: Acha QPPRx. Pembe QPA, APB, na BPR lazima zijumlishe hadi 180circ. Kwa Sheria ya Cosines, angle APBcos-1leftfrac-1124right. Pia, pembe QPA na BPR ni sawa cos-1leftfracx16kulia na cos-1leftfracx 12kulia. Hivyo tunacos-1leftfracx16rightcos-1leftfrac-1124right180 circ-cos-1kushotofracx12kulia.Kuchukua kosini ya pande zote mbili, na kurahisisha kutumia fomula ya nyongeza ya cos na pia utambulisho sin2x cos2x 1, inatoa x 2130. 