This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Haya, twende tukasuluhishe swali hili hatua kwa hatua. a) Urefu wa CB. Tunatumia sheria ya Kosine katika pembetatu BCD. Tunajua BD = 10.9 cm, DC = 64 cm, na BDC = 38^. Step 1: Tumia sheria ya Kosine kupata CB. CB^2 = BD^2 + DC^2 - 2 · BD · DC · ( BDC) CB^2 = (10.9)^2 + (64)^2 - 2 · (10.9) · (64) · (38^) CB^2 = 118.81 + 4096 - 1395.2 · (0.78801) CB^2 = 4214.81 - 1099.50766 CB^2 = 3115.30234 Step 2: Pata thamani ya CB. CB = sqrt(3115.30234) CB ≈ 55.8149 cm Ikiwa imerekebishwa kwa sehemu moja ya desimali: CB ≈ 55.8 cm b) Urefu wa AB. Kwa kuwa ABCD ni trapezi, AB ni sambamba na DC. Hii inamaanisha ABD = BDC (pembe mbadala za ndani). Step 1: Pata ABD na ADB. ABD = BDC = 38^. Katika pembetatu ABD, jumla ya pembe ni 180^. ADB = 180^ - BAD - ABD ADB = 180^ - 45^ - 38^ ADB = 97^ Step 2: Tumia sheria ya Saini katika pembetatu ABD kupata AB. (AB)/(( ADB)) = (BD)/(( BAD)) (AB)/((97^)) = (10.9)/((45^)) AB = (10.9 · (97^))/((45^)) AB = (10.9 · 0.99255)/(0.70711) AB = (10.818795)/(0.70711) AB ≈ 15.2999 cm Ikiwa imerekebishwa kwa sehemu moja ya desimali: AB ≈ 15.3 cm c) Umbali wa wima wa D kutoka AB. Huu ni urefu wa pembetatu ABD kutoka D hadi AB. Step 1: Tumia sheria ya Saini katika pembetatu ABD kupata AD. (AD)/(( ABD)) = (BD)/(( BAD)) (AD)/((38^)) = (10.9)/((45^)) AD = (10.9 · (38^))/((45^)) AD = (10.9 · 0.61566)/(0.70711) AD = (6.709794)/(0.70711) AD ≈ 9.4889 cm Step 2: Pata umbali wa wima h. Umbali wa wima h kutoka D hadi AB ni AD · ( BAD). h = AD · (45^) h = 9.4889 · 0.70711 h ≈ 6.7099 cm Ikiwa imerekebishwa kwa sehemu moja ya desimali: h ≈ 6.7 cm d) Radiasi ya duara inayopita kwenye pointi B, C, na D. Hii ni radiasi ya duara inayozunguka pembetatu BCD (circumradius). Step 1: Pata pembe za pembetatu BCD. Tunajua BD = 10.9 cm, DC = 64 cm, CB ≈ 55.8149 cm, na BDC = 38^. Tumia sheria ya Saini kupata CBD: (DC)/(( CBD)) = (CB)/(( BDC)) ( CBD) = (DC · ( BDC))/(CB) ( CBD) = (64 · (38^))/(55.8149) ( CBD) = (64 · 0.61566)/(55.8149) ( CBD) = (39.40224)/(55.8149) ≈ 0.70593 Kuna uwezekano wa pembe mbili: CBD_1 = (0.70593) ≈ 44.88^ au CBD_2 = 180^ - 44.88^ = 135.12^. Ili kuhakikisha, tunachunguza jumla ya pembe katika pembetatu. Ikiwa CBD = 44.88^, basi BCD = 180^ - 38^ - 44.88^ = 97.12^. Hii inamaanisha upande mfupi zaidi BD (10.9 cm) ungekuwa kinyume na pembe kubwa zaidi (97.12^), ambayo si sahihi. Kwa hivyo, CBD = 135.12^. Sasa, BCD = 180^ - 38^ - 135.12^ = 6.88^. Step 2: Tumia fomula ya radiasi ya duara inayozunguka pembetatu (circumradius) R. R = (CB)/(2 ( BDC)) R = (55.8149)/(2 · (38^)) R = (55.8149)/(2 · 0.61566) R = (55.8149)/(1.23132) R ≈ 45.328 cm Ikiwa imerekebishwa kwa sehemu moja ya desimali: R ≈ 45.3 cm That's 2 down. 3 left today — send the next one.