The diagram below is a quadrilateral ABCD in which CD = 8 cm, BC = 11cm, AC = 13cm, AB = 4cm and angle ADC = 80°.
|Mathematics
The diagram below is a quadrilateral ABCD in which CD = 8 cm, BC = 11cm, AC = 13cm, AB = 4cm and angle ADC = 80°.
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Answer
62.7∘
Haya wmiriam395, twende na swali hili la jiometri.
a) Kokotoa ukubwa wa pembe ACD.
Ili kupata ∠ACD, tutatumia sheria ya Sine katika △ADC.
Tunajua CD=8 cm, AC=13 cm, na ∠ADC=80∘.
Step 1: Tumia sheria ya Sine kupata ∠CAD.
CDsin(∠CAD)=ACsin(∠ADC)8sin(∠CAD)=13sin80∘sin(∠CAD)=138sin80∘sin(∠CAD)=138×0.9848sin(∠CAD)≈0.6060∠CAD=arcsin(0.6060)≈37.3∘
Step 2: Tumia jumla ya pembe katika pembetatu kupata ∠ACD.
Jumla ya pembe katika pembetatu ni 180∘.
∠ACD=180∘−∠ADC−∠CAD∠ACD=180∘−80∘−37.3∘∠ACD=62.7∘
Ukubwa wa pembe ACD ni 62.7∘.
b) Kokotoa urefu wa upande AD.
Tutatumia sheria ya Cosine katika △ADC.
Tunajua CD=8 cm, AC=13 cm, na ∠ADC=80∘.
Step 1: Tumia sheria ya Cosine kupata urefu wa AD.
AC2=AD2+CD2−2(AD)(CD)cos(∠ADC)132=AD2+82−2(AD)(8)cos(80∘)169=AD2+64−16AD(0.1736)169=AD2+64−2.7776AD
Step 2: Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa AD.
AD2−2.7776AD+64−169=0AD2−2.7776AD−105=0
Tumia formula ya quadratic AD=2a−b±b2−4ac:
AD=2(1)2.7776±(−2.7776)2−4(1)(−105)AD=22.7776±7.715+420AD=22.7776±427.715AD=22.7776±20.681
Kwa kuwa urefu hauwezi kuwa hasi, tunachukua thamani chanya:
AD=22.7776+20.681AD=223.4586AD≈11.7293cm
Urefu wa upande AD ni 11.7cm.
c) Kokotoa ukubwa wa pembe ACB.
Tutatumia sheria ya Cosine katika △ABC.
Tunajua AB=4 cm, BC=11 cm, na AC=13 cm.
Step 1: Tumia sheria ya Cosine kupata ∠ACB.
AB2=BC2+AC2−2(BC)(AC)cos(∠ACB)42=112+132−2(11)(13)cos(∠ACB)16=121+169−286cos(∠ACB)16=290−286cos(∠ACB)
Step 2: Tatua kwa cos(∠ACB) na kisha ∠ACB.
286cos(∠ACB)=290−16286cos(∠ACB)=274cos(∠ACB)=286274cos(∠ACB)≈0.9580∠ACB=arccos(0.9580)≈16.7∘
Ukubwa wa pembe ACB ni 16.7∘.
d) Kokotoa eneo la quadrilateral ABCD.
Eneo la quadrilateral ABCD ni jumla ya eneo la △ADC na eneo la △ABC.
Step 1: Kokotoa eneo la △ADC.
Tunajua CD=8 cm, AD≈11.7293 cm, na ∠ADC=80∘.
Eneo(△ADC)=21×CD×AD×sin(∠ADC)Eneo(△ADC)=21×8×11.7293×sin(80∘)Eneo(△ADC)=4×11.7293×0.9848Eneo(△ADC)≈46.20cm2
Step 2: Kokotoa eneo la △ABC.
Tunajua BC=11 cm, AC=13 cm, na ∠ACB≈16.664∘.
Eneo(△ABC)=21×BC×AC×sin(∠ACB)Eneo(△ABC)=21×11×13×sin(16.664∘)Eneo(△ABC)=21×143×0.2866Eneo(△ABC)≈20.49cm2
Step 3: Jumlisha maeneo ya pembetatu mbili.
Eneo(ABCD)=Eneo(△ADC)+Eneo(△ABC)Eneo(ABCD)=46.20+20.49Eneo(ABCD)=66.69cm2
Eneo la quadrilateral ABCD ni 66.7cm2.
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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.
ScanSolve AI Answer
Haya wmiriam395, twende na swali hili la jiometri. a) Kokotoa ukubwa wa pembe ACD. Ili kupata ACD, tutatumia sheria ya Sine katika ADC. Tunajua CD = 8 cm, AC = 13 cm, na ADC = 80^. Step 1: Tumia sheria ya Sine kupata CAD. (( CAD))/(CD) = (( ADC))/(AC) (( CAD))/(8) = ( 80^)/(13) ( CAD) = (8 80^)/(13) ( CAD) = (8 × 0.9848)/(13) ( CAD) ≈ 0.6060 CAD = (0.6060) ≈ 37.3^ Step 2: Tumia jumla ya pembe katika pembetatu kupata ACD. Jumla ya pembe katika pembetatu ni 180^. ACD = 180^ - ADC - CAD ACD = 180^ - 80^ - 37.3^ ACD = 62.7^ Ukubwa wa pembe ACD ni 62.7^. b) Kokotoa urefu wa upande AD. Tutatumia sheria ya Cosine katika ADC. Tunajua CD = 8 cm, AC = 13 cm, na ADC = 80^. Step 1: Tumia sheria ya Cosine kupata urefu wa AD. AC^2 = AD^2 + CD^2 - 2(AD)(CD)( ADC) 13^2 = AD^2 + 8^2 - 2(AD)(8)(80^) 169 = AD^2 + 64 - 16 AD (0.1736) 169 = AD^2 + 64 - 2.7776 AD Step 2: Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa AD. AD^2 - 2.7776 AD + 64 - 169 = 0 AD^2 - 2.7776 AD - 105 = 0 Tumia formula ya quadratic AD = -b ± sqrt(b^2 - 4ac)2a: AD = 2.7776 ± sqrt((-2.7776)^2 - 4(1)(-105))2(1) AD = 2.7776 ± sqrt(7.715 + 420)2 AD = 2.7776 ± sqrt(427.715)2 AD = (2.7776 ± 20.681)/(2) Kwa kuwa urefu hauwezi kuwa hasi, tunachukua thamani chanya: AD = (2.7776 + 20.681)/(2) AD = (23.4586)/(2) AD ≈ 11.7293 cm Urefu wa upande AD ni 11.7 cm. c) Kokotoa ukubwa wa pembe ACB. Tutatumia sheria ya Cosine katika ABC. Tunajua AB = 4 cm, BC = 11 cm, na AC = 13 cm. Step 1: Tumia sheria ya Cosine kupata ACB. AB^2 = BC^2 + AC^2 - 2(BC)(AC)( ACB) 4^2 = 11^2 + 13^2 - 2(11)(13)( ACB) 16 = 121 + 169 - 286 ( ACB) 16 = 290 - 286 ( ACB) Step 2: Tatua kwa ( ACB) na kisha ACB. 286 ( ACB) = 290 - 16 286 ( ACB) = 274 ( ACB) = (274)/(286) ( ACB) ≈ 0.9580 ACB = (0.9580) ≈ 16.7^ Ukubwa wa pembe ACB ni 16.7^. d) Kokotoa eneo la quadrilateral ABCD. Eneo la quadrilateral ABCD ni jumla ya eneo la ADC na eneo la ABC. Step 1: Kokotoa eneo la ADC. Tunajua CD = 8 cm, AD ≈ 11.7293 cm, na ADC = 80^. Eneo( ADC) = (1)/(2) × CD × AD × ( ADC) Eneo( ADC) = (1)/(2) × 8 × 11.7293 × (80^) Eneo( ADC) = 4 × 11.7293 × 0.9848 Eneo( ADC) ≈ 46.20 cm^2 Step 2: Kokotoa eneo la ABC. Tunajua BC = 11 cm, AC = 13 cm, na ACB ≈ 16.664^. Eneo( ABC) = (1)/(2) × BC × AC × ( ACB) Eneo( ABC) = (1)/(2) × 11 × 13 × (16.664^) Eneo( ABC) = (1)/(2) × 143 × 0.2866 Eneo( ABC) ≈ 20.49 cm^2 Step 3: Jumlisha maeneo ya pembetatu mbili. Eneo(ABCD) = Eneo( ADC) + Eneo( ABC) Eneo(ABCD) = 46.20 + 20.49 Eneo(ABCD) = 66.69 cm^2 Eneo la quadrilateral ABCD ni 66.7 cm^2. That's 2 down. 3 left today — send the next one.
