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Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1921452

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Let the square side be a. Calculating a. a²+(0.5a)² = 4² a²+¼(a)² = 16 ¼(5a²) = 16 5a² = 64 a = 8/√(5) a = ⅕(8√(5)) units. Half a is; = ⅕(4√(5)) units. tanb = ((4/√(5))/(8/√(5))) b = atan(0.5)° c = atan(2)° d = (atan(2)-atan(0.5))° tan(atan(2)-atan(0.5)) = e/(4/√(5)) e = (3/√(5)) units. f = a-e f = (8/√(5))-(3/√(5)) f = (5/√(5)) units. It implies; Area yellow (rectangle) is; (8/√(5))*(5/√(5)) = ⅕(40) = 8 square units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1920365

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a = 45-atan(3/4) a = 45-36.86989764584  a = 8.13010235416° Where a is angle DCF. b² = 2(5²) b = 5√(2) cm. Where b is CF = AE. c² = (5√(2))²+4²-10√(2)*4cos8.13010235416   c = 3.16227766017 cm. Where c is DF. (4/sind) = (3.16227766017/sin8.13010235416) d = 10.30484646877° Where d is angle DFC. e = 45-10.30484646877  e = 34.69515353123° Where e is angle AFD. tan34.69515353123 = f/5 f = 3.46153846154 cm. g = 90-34.69515353123 g = 55.30484646877° h = atan(5/(5-3.46153846154)) h = 72.89727103...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1919801

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Let the regular pentagon side be 1 unit. sin72 = a/1 a = 0.9510565163 unit. Let the side of the two inscribed congruent squares be b. c = ½(b) unit. b+½(b) = 0.9510565163 b = 0.63403767753 unit. Area Yellow is; (0.63403767753)² = 0.40200377653 square units. Calculating Area Blue. d = ½(1-0.63403767753) d = 0.18298116124 unit. e² = 2-2cos108 e = 1.61803398875 units. f = ½(1.61803398875-0.63403767753) f = 0.49199815561 unit. Area Blue is; 0.5(0.49199815561+0.18298116124)0.9510565163 =...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1919754

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Let the longer diagonal be 1 unit. (1/sin150) = (a/sin18) a = 0.61803398875 units. b = 12° (1/sin150) = (c/sin12) c = 0.41582338164 units. d = 180-84-42 d = 54° (e/sin54) = (0.41582338164/sin42) e = 0.50275413979 units. f² = 0.50275413979²+0.61803398875²-2*0.61803398875*0.50275413979cos66 f = 0.61803398875 units. (0.61803398875/sin66) = (0.50275413979/sing) g = 48° It implies; The required angle, theta is; 48-b = 48-12 = 36°

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1919356

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Notice; Triangle ABC is equilateral. Let the side of the regular pentagon be a. Single interior angle of the regular pentagon is; ⅕(180*3) = 108° (HF)² = 2a²-2a²cos108 HF = 1.61803398875a units. sin18 = b/a b = 0.30901699437a units. c = HF-b c = 1.61803398875a-0.30901699437a c = 1.30901699438a units. d = 1-c d = (1-1.30901699438a) units. cos18 =e/a e = 0.9510565163a units. Calculating a, side length of the regular pentagon. tan60 = e/d √(3) = 0.9510565163a/(1-1.30901699438a) √(3)-2.26...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1919162

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Let AB be 1 unit. (1/sin40) = (BC/sin70) BC = 1.46190220008 units. (1.46190220008/sin120) = (AE/sin40) AE = 1.08506357513 units. (1.08506357513/sin40) = (CE/sin20) CE = 0.57735026919 units. BE = 1.46190220008-0.57735026919 BE = 0.88455193089 units. (BD)² = 0.88455193089²+0.57735026919²-2*0.57735026919*0.88455193089cos60 BD = 0.77786191343 units. Therefore angle alpha (angle BDE) is; Let it be x. (0.88455193089/sinx) = (0.77786191343/sin60) x = 80° Angle alpha (required angle) = 80°

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1918922

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a² = 12²-6² a = 6√(3) cm. Where a is BD. b = ½(a) b = 3√(3) cm. Where b is BG = DG. c = atan(6/(3√(3)) c = 49.10660535087° Where c is angle AGD = angle BGE. d = 60-49.10660535087 d = 10.89339464913° Where d is angle BAE. e = 180-60-10.89339464913 e = 109.10660535087° Where e is angle BEA. (12/sin109.10660535087) = (f/sin60) f = 10.99818166789 cm. Where f is AE. (12/sin109.10660535087) = (g/sin10.89339464913) g = 2.4 cm. Where g is BE. h = 12-2.4 h = 9.6 cm. Where h is CE. i² = (6√(3))²...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1916952

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Let AB be 5 units. Radius, r of the inscribed circle is; 5*(2/5) r = 2 units. a = ⅛(180*6) a = 135° 2b² = 25 b = ½(5√(2)) units. c = 180-0.5(135)-45 c = 67.5° sin22.5 = ½(5√(2))/d d = 9.23879532511 units. Where d is the side length of the regular triangle (purple area). Area purple is; 0.5*9.23879532511*9.23879532511sin60 = 36.95994598698 square units. Calculating Area Shaded. e = 360-135-45-67.5-60 e = 52.5° f² = 50-50cos135 f = 9.23879532511 units. g = 52.5+0.5(45) g = 75° sin(...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1916500

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AD = √(40²-20²) AD = 20√(3) units. AE = DE = ½(AD) = 10√(3) units. Angle BED is; = atan(20/(10√(3))) = atan(⅔√(3))° = 49.10660535087° Angle DAC is; = 30° (EF)² = 300+625-20√(3)*25cos30 EF = 5√(7) units. (5√(7)/sin30) = (25/sina) a = 70.89339464913° a is angle DEF. Therefore required angle is; Angle BED + Angle DEF = 49.10660535087+70.89339464913 = 120°

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1916431

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Let AD = 2AB. For AB = 2 units, AD = BC = 4 units. Angle APD is; = atan(4)° PD = √(4²+1)  PD = √(17) units. Considering similar triangles APD and QCD ratios. 1 = a √(17) = 2 Crossing Multiply. a = (2/√(17)) units. a = 0.48507125007 units. a is DQ. PQ = √(17)-0.48507125007  PQ = 3.63803437554 units. Angle BPQ = 180-atan(4) = 104.03624346793° Calculating QB. Notice, BC is 4 units. (QB)² = 3.63803437554²+1-2*3.63803437554cos104.03624346793 QB = 4 unit. It implies;  BC = QB Proved....
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