<prompt> Mstatili ABCD na nusu duara yenye kipenyo AB ni coplanar na hazina mambo ya ndani yasiyopishana. Acha mathcalR iashiria eneo lililofungwa na nusu duara na mstatili. Mstari ell hukutana na nusu duara, sehemu AB, na sehemu CD katika pointi tofauti N, U, na T, mtawalia. Laini ya ell inagawanya eneo mathcalR katika maeneo mawili yenye maeneo katika uwiano wa 1:2. Tuseme kwamba AU 84, AN 126, na UB 168. Kisha DA inaweza kuwakilishwa kama msqrt n, ambapo m na n ni nambari kamili chanya na n haiwezi kugawanywa kwa mraba wa mkuu wowote. Tafuta m n.asy kuagiza grafu; defaultpenlinewidth0.7fontsize10; ukubwa 500; kalamu zzttqq rgb0.6,0.2,0; kalamu xdxdff rgb0.4902,0.4902,1; sehemu na takwimu chora0,-154.31785--0,0; chora0,0--252,0; chora0,0--126,0,zzttqq; kuchora126,0--63,109.1192,zzttqq; kuchora63,109.1192--0,0,zzttqq; kuchora-71.4052,9166.01287-109.1192-71.405221--504.60925,9166.01287-109.1192504.6092521; kuchora0,-154.31785--252,-154.31785; kuchora252,-154.31785--252,0; chora0,0--84,0; chora84,0--252,0; kuchora63,109.1192--63,0; kuchora84,0--84,-154.31785; choraarc126,0,126,0,180; pointi na lebo nukta 0,0; leboA,-16.43287,-9.3374,NE2; nukta 252,0; leboB,255.242,5.00321,NE2; nukta 0,-154.31785; leboD,3.48464,-149.55669,NE2; nukta 252,-154.31785; leboC,255.242,-149.55669,NE2; nukta 126,0; leboO,129.36332,5.00321,NE2; nukta 63,109.1192; leboN,44.91307,108.57427,NE2; lebo126,28.18236,40.85473,NE2; nukta 84,0; leboU,87.13819,5.00321,NE2; nukta 113.69848,-154.31785; leboT,116.61611,-149.55669,NE2; nukta 63,0; leboN,66.42398,5.00321,NE2; lebo84,41.72627,-12.5242,NE2; lebo168,167.60494,-12.5242,NE2; nukta 84,-154.31785; leboT,87.13819,-149.55669,NE2; nukta 252,0; leboI,255.242,5.00321,NE2; klipu-71.4052,-225.24323---71.4052,171.51361--504.60925,171.51361--504.60925,-225.24323--mzunguko; asy  <response>: Katikati ya semicircle pia ni katikati ya AB. Acha hatua hii iwe O. Acha h iwe urefu wa AD.Rejesha kila kitu kwa 42, hivyo AU 2, AN 3, UB 4. Kisha AB 6 hivyo OA OB 3.Kwa kuwa ON ni kipenyo cha nusu duara, ON 3. Kwa hivyo OAN ni pembetatu iliyo sawa.Acha X, Y, na Z ziwe maeneo ya pembetatu OUN, sekta ONB, na trapezoid UBCT mtawalia.X frac 12UONOsinO frac 1213sin60circ frac 3 4sqrt 3Y frac 13pi32 3piIli kupata Z inabidi tutafute urefu wa TC. Mradi T na N kwenye AB ili kupata pointi T na N. Ona kwamba UNN na TUT zinafanana. Hivyo:frac TTUT frac UNNN inamaanisha frac TTh frac 123sqrt 32 inamaanisha TT frac sqrt 39h.Kisha TC TC - TT UB - TT 4 - frac sqrt 39h. Kwa hivyo:Z frac 12BU TCCB frac 12left8 - frac sqrt 39hrighth 4h - frac sqrt 318h2Acha L liwe eneo la upande wa mstari l ulio na mikoa X, Y, Z. KishaL X Y Z frac 34sqrt 3 3pi 4h - frac sqrt 318h2Ni wazi, L ni kubwa kuliko eneo la upande mwingine wa mstari l. Eneo hili lingine ni sawa na jumla ya eneo kasoro L. Hivyo:frac 21 frac L6h frac 92pi - L inamaanisha 12h 9pi 3L.Sasa suluhisha kwa h.anzaalign 12h 9pi frac 94sqrt 3 9pi 12h - frac sqrt 36h2 0 frac 94sqrt 3 - frac sqrt 36h2 h2 frac 946 h frac 32sqrt 6 endalignUsisahau kuondoa ukubwa mwishoni ili kupata AD frac 32sqrt 6 cdot 42 63sqrt 6.Hatimaye, jibu ni 63 6 69. 