Mathematics Question and Solution

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Calculating Length x (AD).Let a be the diameter of the green inscribed circle.Therefore;x ~ a(b+x) ~ 2aCross Multiply.2ax = ab+ax2ax-ax = abax = abx = b units.Therefore, length x is;x*x = 9*4x² = 36x = √(36)x = 6 units.

Mathematics Question and Solution

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Calculating Inscribed Square Area.Notice.The ascribed right-angled triangle is not drawn to scale.Let a be the side length of the inscribed square.Calculating Area Inscribed Square.tanb = 3√(3)/(9√(3))b = atan(⅓)°tanb = a/ctan(atan(⅓)) = a/cc = 3a units.d = 9√(3)-c-ad = (9√(3)-4a) units.e = (3√(3)-a) units.Calculating a.(3√(3))² = d²+e²27 = (9√(3)-4a)²+(3√(3)-a)²27 = 243-72√(3)a+16a²+27-6√(3)a+a²17a²-78√(3)a+243 = 0a²-(78√(3)/(2*17))a = -243/17(a-(39√(3)/17))² = (-243/17)+(-39√(3)/17))²(a-(39√(...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3557803

Calculating x.Notice.The angle formed at the meet point (vertex) of length 4 units and x units is equal the angle formed at the meet point (vertex) 9 units and x units.It implies, two similar plane shapes, quadrilaterals are observed.Calculating x.x ~ 94 ~ xCross Multiply.x² = (9*4)x = √(36)x = 6 units.

Mathematics Question and Solution

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Calculating Shaded Fraction.Let the bigger inscribed semi circle radius be 2 units.Let the big inscribed semi circle radius be r.Let the inscribed circle radius be q.Calculating r.(2+r)² = 2²+(4-r)²4+4r+r² = 4+16-8r+r²4r = 16-8r12r = 163r = 4r = 4/3 units.Calculating q.(2+(4/3))² = (2+q)²+((4/3)+q)²(10/3)² = 4+4q+q²+(16/9)+(8q/3)+q²(100/9)-(16/9)-4 = 2q²+4q+(8q/3)(84/9)-4 = 2q²+4q+(8q/3)84-36 = 18q²+36q+24q48 = 18q²+60q6q²+20q-16 = 03q²+10q-8 = 0Therefore;q ≠ -4q = ⅔ units.Blue shaded area is;A...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3555938

Calculating Length EF.Let FC = x.Let CE = y.Therefore;5+x = y+4x = y-1 --- (1).Notice;Triangle AEF is similar to triangleEFC.It implies;5+x ~ y 4 ~ xCross Multiply.x²+5x = 4y --- (2).Substituting (1) in (2).(y-1)²+5(y-1) = 4yy²-2y+1+5y-5 = 4yy²-y-4 = 0(y-½)² = 4+(¼)y = ½±½√(17)Therefore;y ≠ ½-½√(17)y = ½+½√(17)y = ½(√(17)+1)Therefore;Calculating x.At (1).x = y-1, and y = ½(√(17)+1)x = ½(√(17)+1)-1x = ½(√(17)-1)Therefore length EF is;√(y²+x²)= √((½(√(17)+1))²+(½(√(17)-1))= √(¼(34+2))= √(¼(36)...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3555922

Calculating Theta.Let it be x Let 1 units be the side length of the square.tan15 = b/1b = tan15 units.b = 0.26794919243 units.b is length AF.a+b = 1a = 1-ba = 1-0.26794919243a = 0.73205080757 units.a is length AE = length DF.c² = 1²+0.73205080757²-2*1*0.73205080757cos30c = 0.5176380902 units.c is length DE.Calculating Theta, angle x.It implies;(0.73205080757/sinx) = (0.5176380902/sin30)sinx = (0.73205080757sin30)/0.5176380902x = asin(0.70710678119)x = 45°Again, x is theta.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3555147

Calculating x/y please.Notice.The side length of the regular decagon and regular pentagon are equal.Let 2 units be the side length of the regular decagon.Calculating x.a = ⅒*180(10-2)a = 144°a is the single interior angle of the regular decagon.b = ⅕*180(5-2)b = 108°b is the single interior angle of the regular pentagon.c = 180-108c = 72°d = ½(c)d = 36°e = 144-c-de = 144-108e = 36°f² = 2²+2²-2*2*2cos108f = 3.2360679775 units.g² = 3.2360679775²+2²-2*2*3.2360679775cos36g = 2 units.h = 144-108h =...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3554768

Calculating R, radius of the circle.a = (R-6) units.b = (6+R) units.It implies;b*a = 3*7(6+R)(R-6) = 216R-36+R²-6R = 21R²-36 = 21R² = 21+36R² = 57R = √(57) units.R = 7.54983443527 units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3554214

Notice.R-r = 10 cm.It implies;R = (10+r) cm.R is the side length of the ascribed square.Calculating length AB.a = R-ra = 10 cm.b = ½(R) units.It implies;R² = 10²+(½(R))²R²-¼(R²) = 100¼(3R²) = 1003R² = 400R = √(400/3)R = ⅓(20√(3)) cm.R = 11.5470053838 cm.And R-r = 10 cm.Therefore;⅓(20√(3))-10 = rr = ⅓(20√(3)-30) cm.r = 1.5470053838 cm.Recall.b = ½(R) units.b = ½(11.5470053838)b = 5.7735026919 cm.tanc = 10/5.7735026919c = 60°d = 90-cd = 30°tan(0.5*30) = e/11.5470053838e = 11.5470053838tan(15)e =...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3553344

Calculating Length Red.Notice.Radius of the half circle is 5 units.a = (5-x) units.Calculating x.5² = 3²+(5-x)²16 = 25-10x+x²x²-10x+9 = 0Therefore;x ≠ 9 units.x = 1 unit.Recall.a = 5-xAnd x = 1 unit.It implies;a = 4 units.Therefore, length red is;Let it be y.y² = (5+3)²+a²And a = 4 units.y² = 8²+4²y = √(64+16)y = √(80)y = 4√(5) units.

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