Let CD be b units.
Let BC be a units.
c = 2+4
c = 6 units.
Observing similar plane shape (right-angled) side length ratios.
a - 6
6 - b
Cross Multiply.
ab = 36 --- (1).
Let alph...
tana = (0.5*4)/6
tana = ⅓
a = atan(⅓)°
b = 2a
b = 2atan(⅓)°
c = ½(180-b)
c = ½(180-36.8698976458)
c = 71.5650511771°
d = c+½(180-c)
d = 125.7825255886°
e = 180-d
e = 54.2174744115°
f = 180-125....
Let a be the side length of the inscribed blue square.
b² = 2a²
b = √(2)a units.
2c² = a²
c = √(a²/2)
c = ½√(2)a unit.
It implies;
1² = (√(2)a)²+(½√(2)a)²
1 = 2a²+½(a²)
1 = ½(5a²)...
For y = (25/16) unit.
x is;
(25/16) = (5/4)^(2x)
2x = 2
x = 1 unit.
Gradient of the curve at x equal 1 unit is;
dy/dx = (25/8)ln(5/4)
At y = 0 at the tangent, x is;
x = (50ln(5/4)-25)/50ln(5/4)...
Radius of the inscribed circle, r is;
√(5)r + r = 2
r = 2/(√(5)+1) unit.
r = ½(√(5)-1) unit
Therefore;
Area Yellow exactly is square units is;
2(area triangle with height (√(5)-1) unit...
Sir Mike Ambrose is the author of the question.
tan30 = a/6
a = 2√(3) cm.
b = (6-2√(3)) cm.
sin30 = 2√(3)/c
c = 4√(3) cm.
6d+2√(3)d+ 4√(3)d = 6*2√(3)
(6+6√(3))d = 12√(3)
d = (72√(3)-216...
Showing that quadrilateral ABCD is cyclic.
Notice!
For quadrilateral ABCD to be cyclic;
Angle ADC will equal 2 times angle ABC.
Angle ADC = 2(Angle ABC)
It implies;
Since Angle ABC...
Blue area is;
½(9x2)+½(√(68)x√(17))
= 9+½(34)
= 9+17
= 26 cm²
RT = √((8+5)²-5²)
RT = √(169-25)
RT = √(144)
RT = 12 cm.
RT is the required length.
Blue area will be;
½(5.2x6.4)+½(2x8)
= ½(33.28)+8
= 16.64+8.00
= 26.64 square unit.
