area of shaded region = area of sector ACD - area of blue segment - area of △ACD
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AB and DB are both radiuses of circle B so they are the same length.
AD = DB = AB = 20
So △ABD is an equilateral triangle.
m∠ABD = 60°
m∠ACD = 2( m∠ABD ) = 2( 60° ) = 120°
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m∠GCD = (1/2)(m∠ACD) = (1/2)(120°) = 60°
△CGD is a 30-60-90 triangle.
GD = (1/2)(AD) = (1/2)(20) = 10
CG = 10 / sqrt(3)
CD = 20 / sqrt(3)
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area of sector ABD = \(\pi\cdot(\text{radius})^2\cdot\big(\frac{\text{angle}}{360^\circ}\big)\ =\ \pi\cdot(20)^2\cdot\big(\frac{60^\circ}{360^\circ}\big)\ =\ \frac{400\pi}{6}\)
area of △ABD = \(\frac12\cdot(\text{base})\cdot(\text{height})\ =\ \frac12\cdot(20)\cdot(10\sqrt3)\ =\ 100\sqrt3\)
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area of blue segment = area of sector ABD - area of △ABD = \(\frac{400\pi}{6}-100\sqrt3\)
area of sector ACD = \(\pi\cdot(\text{radius})^2\cdot\big(\frac{\text{angle}}{360^\circ}\big)\ =\ \pi\cdot\Big(\frac{20}{\sqrt3}\Big)^2\cdot\Big(\frac{120^\circ}{360^\circ}\Big)\ =\ \frac{400\pi}{9}\)
area of △ACD = \(\frac12\cdot(\text{base})\cdot(\text{height})\ =\ \frac12\cdot(20)\cdot(\frac{10}{\sqrt3})\ =\ \frac{100}{\sqrt3}\)
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area of shaded region = \(\frac{400\pi}{9}-(\frac{400\pi}{6}-100\sqrt3)-\frac{100}{\sqrt3}\ \approx\ 46\) square units
area of shaded region = area of sector ACD - area of blue segment - area of △ACD
----------
AB and DB are both radiuses of circle B so they are the same length.
AD = DB = AB = 20
So △ABD is an equilateral triangle.
m∠ABD = 60°
m∠ACD = 2( m∠ABD ) = 2( 60° ) = 120°
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m∠GCD = (1/2)(m∠ACD) = (1/2)(120°) = 60°
△CGD is a 30-60-90 triangle.
GD = (1/2)(AD) = (1/2)(20) = 10
CG = 10 / sqrt(3)
CD = 20 / sqrt(3)
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area of sector ABD = \(\pi\cdot(\text{radius})^2\cdot\big(\frac{\text{angle}}{360^\circ}\big)\ =\ \pi\cdot(20)^2\cdot\big(\frac{60^\circ}{360^\circ}\big)\ =\ \frac{400\pi}{6}\)
area of △ABD = \(\frac12\cdot(\text{base})\cdot(\text{height})\ =\ \frac12\cdot(20)\cdot(10\sqrt3)\ =\ 100\sqrt3\)
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area of blue segment = area of sector ABD - area of △ABD = \(\frac{400\pi}{6}-100\sqrt3\)
area of sector ACD = \(\pi\cdot(\text{radius})^2\cdot\big(\frac{\text{angle}}{360^\circ}\big)\ =\ \pi\cdot\Big(\frac{20}{\sqrt3}\Big)^2\cdot\Big(\frac{120^\circ}{360^\circ}\Big)\ =\ \frac{400\pi}{9}\)
area of △ACD = \(\frac12\cdot(\text{base})\cdot(\text{height})\ =\ \frac12\cdot(20)\cdot(\frac{10}{\sqrt3})\ =\ \frac{100}{\sqrt3}\)
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area of shaded region = \(\frac{400\pi}{9}-(\frac{400\pi}{6}-100\sqrt3)-\frac{100}{\sqrt3}\ \approx\ 46\) square units
It appears that the chord AD is bisected by BE
So angle BED is right and ED = 1/2 AD = 10
And we can find angle EBD using the Law of Sines
sin BED sin (EBD)
_______ = __________
BD ED
sin 90 sinEBD
______ = ______
20 10
sinEBD = (10/20) sin 90
sin EBD = (1/2) (1)
sin EBD = 1/2
So....EBD must be 30°
So.....by symmetry....angle ABD must be twice this = 60°
So....now....we can compute the area of sector AEBD =
pi* BD^2(60/360) = pi* (20)^2 (1/6) = pi * 400/6 = 200/3 pi units^2
And since angle ACD is twice angle ABD.....then (1/2)ACD = ABD = 60° = angle ECD
Then....since DEC is right, then triangle ECD is a 30-60-90 right triangle
Then
CD / sin DEC = ED / sin ECD
CD / 1 = 10/ [sqrt(3)/2 ]
CD = 20 /sqrt (3)
And the area of sector AFDC =
pi (CD^2) (120/360) = pi * (400/3)(1/3) = (400/9) pi units^2 (1)
Now....we need to caclculate the area of triangle ABD
(1/2)(AB)(BD) sin (60°) =
(1/2)(20)(20)sqrt (3)/2) = 100sqrt(3) units^2
So.....the area between chord AD and arc AED =
Area of AEBD - area of triangle ABD =
( [200/3] pi - 100sqrt(3) ) (2)
Lastly....we need to calculate the area of triangle ACD [note CD = AC ] =
(1/2)(CD)(AC)sin(120°) = (1/2)(20/sqrt3)^2 sqrt(3)/2 = (1/2)(400/3)sqrt(3)/2 = 100/sqrt(3) = 100sqrt(3)/3 (3)
So....the area we are looking for is :
Area of sector AFDC - area between chord AD and arc AED - area of triangle ACD =
(1) - (2) - (3) =
[400/9] pi - [ 200/3 ]pi + 100sqrt(3) - 100sqrt(3)/3=
[ 400/9]pi - [600/9]pi + (2/3)100sqrt(3) =
(2/3)100sqrt(3) - (200/9)pi ≈ 45.66 units^2 = 46 units^2
[If I didn't make any errors....LOL!!! ]