Construct a trapezium ABCD in which AB = 10cm, BC = 8cm, AD = 7cm, angle ABC = 60° and angle BAD is 90°.

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 $10 \text{ cm}$.
2. Kwenye ncha $A$, jenga pembe ya $90^\circ$ ($\angle BAD$).
3. Kwenye mstari wa pembe ya $90^\circ$ kutoka $A$, pima na uweke alama $D$ ili $AD = 7 \text{ cm}$.
4. Kwenye ncha $B$, jenga pembe ya $60^\circ$ ($\angle ABC$).
5. Kwenye mstari wa pembe ya $60^\circ$ kutoka $B$, pima na uweke alama $C$ ili $BC = 8 \text{ cm}$.
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 $\angle BAD = 90^\circ$ na $AD = 7 \text{ cm}$, kuratibu za $D$ ni $(0,7)$.
Kwa $C$, tunajua $BC = 8 \text{ cm}$ na $\angle ABC = 60^\circ$.
Kuratibu za $C$ zinaweza kupatikana kwa kutumia trigonometry kutoka $B$:

$$x_C = x_B + BC \cos(180^\circ - 60^\circ) = 10 + 8 \cos(120^\circ) = 10 + 8(-\frac{1}{2}) = 10 - 4 = 6$$ $$y_C = y_B + BC \sin(180^\circ - 60^\circ) = 0 + 8 \sin(120^\circ) = 8(\frac{\sqrt{3}}{2}) = 4\sqrt{3}$$

Hivyo, $C = (6, 4\sqrt{3})$.
Sasa, tunaweza kutumia formula ya umbali kati ya $D(0,7)$ na $C(6, 4\sqrt{3})$:

$$CD = \sqrt{(x_C - x_D)^2 + (y_C - y_D)^2}$$ $$CD = \sqrt{(6 - 0)^2 + (4\sqrt{3} - 7)^2}$$ $$CD = \sqrt{6^2 + (4\sqrt{3})^2 - 2(4\sqrt{3})(7) + 7^2}$$ $$CD = \sqrt{36 + (16 \times 3) - 56\sqrt{3} + 49}$$ $$CD = \sqrt{36 + 48 - 56\sqrt{3} + 49}$$ $$CD = \sqrt{133 - 56\sqrt{3}}$$

Kukokotoa thamani ya namba:

$$CD \approx \sqrt{133 - 56 \times 1.73205}$$ $$CD \approx \sqrt{133 - 96.9948}$$ $$CD \approx \sqrt{36.0052}$$ $$CD \approx 6.00 \text{ cm}$$

Urefu wa $CD$ ni takriban $\boxed{\text{6.0 cm}}$.

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 $2 \text{ cm}$ kutoka $CD$, na ndani ya trapezi $ABCD$.
• (ii) 9cm from B: Hii ni sehemu ya duara (arc) yenye kitovu $B$ na rediasi $9 \text{ cm}$, ndani ya trapezi $ABCD$.
• (iii) equidistant from AB and AD: Hii ni mstari unaogawanya pembe $BAD$ katika sehemu mbili sawa (angle bisector). Kwa kuwa $\angle BAD = 90^\circ$, mstari huu utakuwa kwenye $45^\circ$ 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.