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 la 4. Kwa kuwa mimi ni AI, siwezi kuchora au kufanya ujenzi halisi. Hata hivyo, nitaeleza hatua za ujenzi na maelezo ya loci kama inavyohitajika. 4(a) Construct a trapezium ABCD Hatua za ujenzi: 1. Chora mstari AB wenye urefu wa 10 cm. 2. Kwenye ncha A, jenga pembe ya 90^ ( BAD). 3. Kwenye mstari wa pembe ya 90^ kutoka A, pima na uweke alama D ili AD = 7 cm. 4. Kwenye ncha B, jenga pembe ya 60^ ( ABC). 5. Kwenye mstari wa pembe ya 60^ kutoka B, pima na uweke alama C ili BC = 8 cm. 6. Unganisha ncha C na D kukamilisha trapezi. 4(b) Measure and write the length of CD. Ili kupata urefu wa CD bila kuchora, tunaweza kutumia jiometria ya kuratibu. Tuchukulie A iko kwenye asili (0,0). Hivyo, A = (0,0) na B = (10,0). Kwa kuwa BAD = 90^ na AD = 7 cm, kuratibu za D ni (0,7). Kwa C, tunajua BC = 8 cm na ABC = 60^. Kuratibu za C zinaweza kupatikana kwa kutumia trigonometry kutoka B: x_C = x_B + BC (180^ - 60^) = 10 + 8 (120^) = 10 + 8(-(1)/(2)) = 10 - 4 = 6 y_C = y_B + BC (180^ - 60^) = 0 + 8 (120^) = 8(sqrt(3)2) = 4sqrt(3) Hivyo, C = (6, 4sqrt(3)). Sasa, tunaweza kutumia formula ya umbali kati ya D(0,7) na C(6, 4sqrt(3)): CD = sqrt((x_C - x_D)^2 + (y_C - y_D)^2) CD = sqrt((6 - 0)^2 + (43) - 7)^2 CD = sqrt(6^2 + (43))^2 - 2(4sqrt(3))(7) + 7^2 CD = sqrt(36 + (16 × 3) - 563) + 49 CD = sqrt(36 + 48 - 563) + 49 CD = sqrt(133 - 563) Kukokotoa thamani ya namba: CD ≈ sqrt(133 - 56 × 1.73205) CD ≈ sqrt(133 - 96.9948) CD ≈ sqrt(36.0052) CD ≈ 6.00 cm Urefu wa CD ni takriban 6.0 cm. 4(c) Within the trapezium ABCD, draw the locus of points which are (i) 2cm from CD: Hii ni mstari sambamba na CD, umbali wa 2 cm kutoka CD, na ndani ya trapezi ABCD. (ii) 9cm from B: Hii ni sehemu ya duara (arc) yenye kitovu B na rediasi 9 cm, ndani ya trapezi ABCD. (iii) equidistant from AB and AD: Hii ni mstari unaogawanya pembe BAD katika sehemu mbili sawa (angle bisector). Kwa kuwa BAD = 90^, mstari huu utakuwa kwenye 45^ kutoka AB (na AD), ukianzia A, na ndani ya trapezi ABCD. 4(d) A point E within the trapezium ABCD, is such that it is greater than or equal to 2cm from CD, less than or equal to 9cm from B and nearer to AD than AB. Indicate, by shading, the region in which E must lie. Eneo ambalo E lazima liwe ndani yake linafafanuliwa na masharti matatu: 1. Greater than or equal to 2cm from CD: Hii inamaanisha E iko kwenye upande wa mstari wa locus kutoka 4(c)(i) ambao uko mbali zaidi na CD. 2. Less than or equal to 9cm from B: Hii inamaanisha E iko ndani au kwenye arc ya locus kutoka 4(c)(ii). 3. Nearer to AD than AB: Hii inamaanisha E iko kwenye upande wa mstari wa locus kutoka 4(c)(iii) ambao uko karibu zaidi na AD. Ili kuonyesha eneo hili kwa kivuli, ungetakiwa kuchora loci zote tatu kwanza. Kisha, eneo la kivuli litakuwa makutano ya maeneo yote matatu yanayokidhi masharti yaliyotajwa hapo juu. Uko vizuri! Tuma swali linalofuata 📸