Calculating the inscribed orange triangle area.
Notice.
a = ½(3)
a = 1.5 units.
a is the radius of the inscribed circle.
tanb = 3/4
b = atan(3/4)°
tan(atan(¾)) = c/1.5
¾ = ⅓(2c)
c...
Calculating x.
a = (x+1) units.
b² = 1²+1²
b² = 2
b = √(2) units.
c = b+r
c = (1+√(2)) units.
Therefore, x is;
(x+1)² = x²+(1+√(2))²
x²+2x+1 = x²+1+2√(2)+2
2x = 2√(2)+2
2x...
Calculating k.
k^(k) = 3^(k)*3⁹
k^(k) = 3^(k)*19683
(k/3)^(k) = 19683
It implies;
k = 9
Checking Accuracy.
(9/3)⁹ = 3⁹ = 19683
Calculating x, the side length of the square ABCD.
3² = x²+4²-2*4xcosy
9 = x²+16-8xcosy
8xcosy = x²+7
cosy = (x²+7)/(8x) --- (1).
5² = x²+4²-2*x*4cos(90-y)
25 = x²+16-8xcos(90-y)
8x(cos9...
Calculating Shaded Area.
a²+10² = 12²
a = √(144-100)
a = √(44)
a = 2√(10) units.
b = 10-a
b = (10-2√(10)) units.
b = 3.67544467966 units.
cosc = 10/12
c = acos(⅚)°
d = 90-45-c
d...
Calculating Yellow Area.
Area yellow is;
Area triangle with two side 8√(2) cm and 2 cm, and angle 45° - Area triangle with height (20/3) cm and base ⅓(20tan(45-2atan(⅓))) cm
= ½*2*8√(2)sin45 - ½*...
Calculating Blue Area.
Let a be the congruent inscribed side lengths each of the inscribed right-angled isosceles triangle.
b = ½(2)
b = 1 unit.
c²+1² = a²
c = √(a²-1) units.
d = ½(18...
Calculating Area Purple ÷ Area Orange Exactly.
Area purple is;
Area trapezoid with two parallel sides 4 cm and (50/9) cm, and height (7/2) cm - Area triangle with height 4 cm and base 3 cm....
Calculating angle alpha, let it be x.
Let R = 2 units.
a = (2-r) units.
Calculating r.
It implies;
r²+r² = (2-r)²
2r² = 4-4r+r²
r²+4r-4 = 0
(r+2)² = 4+(2)²
r = -2±√(8)...
Calculating the required angle, let it be x.
Let 1 unit be the side length of the both congruent inscribed equilateral triangles.
a = ⅑*180(9-2)
a = 140°
a is the single interior angle of t...
