<prompt> Umbali kutoka kwa uhakika P hadi tano ya wima ya octahedron ya kawaida ni 3, 7, 8, 9, na 11. Tafuta umbali kutoka P hadi vertex ya sita.asykuagiza tatu;ukubwa 125;currentprojection mtazamo 6,3,1;mara tatu A, B, C, D, E, F, P;A 1,0,0;B -1,0,0;C 0,1,0;D 0,-1,0;E 0,0,1;F 0,0,-1;P 1.2,1.5,1;kuchoraA--P,nyekundu;choraB--P,nyekundu;kuchoraC--P,nyekundu;kuchoraD--P,nyekundu;kuchoraE--P,nyekundu;choraF--P,nyekundu;kuchoraA--C;kuchoraA--D;kuchoraA--E;kuchoraA--F;kuchoraC--E--D--F--mzunguko;kuchoraD--B--C,dashed;kuchoraB--C,dashed;kuchoraB--D,dashed;kuchoraB--E,dashed;kuchoraB--F,dashed;leboP, P, NE;asy  <response>: Acha P x,y,z, na wima za oktahedron ziwe A a,0,0, B -a,0,0, C 0,a,0, D 0,-a,0, E 0,0,a, na F 0,0,-a. Kisha miraba ya umbali kutoka P hadi wima nianzaalignd_A2 x - a2 y2 z2, d_B2 x a2 y2 z2, d_C2 x2 y - a2 z2, d_D2 x2 y a2 z2, d_E2 x2 y2 z - a2, d_F2 x2 y2 z a2.endalignKumbuka hilod_A2 d_B2 d_C2 d_D2 d_E2 d_F2 2x2 2y2 2z2 2a2.Kati ya masafa 3, 7 , 8, 9, na 11, tunaangalia jumla ya miraba yao katika jozi:anzaalign32 72 58, 32 82 73, 32 92 90, 32 112 130, 72 82 113, 72 92 130, 72 112 170, 82 92 145, 82 112 185, 92 112 202.endalignTunaona thamani moja pekee iliyorudiwa, yaani 32 112 72 92 130. Kwa hivyo, umbali wa sita lazima uwe sqrt130 - 82 sqrt66. 