+128633 First one
Triangle FCB similar to Triangle DCH
HD / DC = FB / FC
5 / DC = FB / FC (1)
Triangle BDA similar to Triangle BFC
FB/ FC = BD / DA
FB /FC = BD / 16 (2)
Equate (1) , (2)
5 / DC = BD / 16
80 = DC * BD { DC = BD}
80 = DC * DC
DC^2 = 80
DC = 4sqrt 5
[ ABC ] = DC * AD = 4sqrt (5) * 16 = 64sqrt (5)
+128633 Second one
Law of Cosines
BC^2 = 4^2 + 5^2 - 2 (4*5)cos 60°
BC^2 = 41 - 20
BC = sqrt (21)
Law of Cosines again
4^2 = 5^2 + 21 - 2(5 sqrt 21) cos BCD
[16 - 25 - 21] / [-2 * 5 *sqrt 21] =cos BCD
[ -30] / [ -10 sqrt 21 ] = cos BCD
3/sqrt21 = cos BCD = sin ACE .......cos ACE = sqrt 12 / sqrt 21
Angle ACE + Angle CAE + Angle AEC =180
Angle ACE + Angle CAE + 60 = 180
Angle ACE + Angle CAE = 120
Angle CAE = 120 - Angle ACE
sin (angle CAE) = sin (120 - angle ACE) =
sin (120)cos ACE - cos (120)sin ACE =
sqrt 3 / 2 * sqrt 12 /sqrt 21 + 1 / 2 * 3/sqrt 21 =
3/sqrt 21 + (3/2)/sqrt 21 =
4.5 /sqrt 21
Law of Sines
CE / sin CAE = AE / sin ACE
4 / (4.5/sqrt21) = AE / (3/sqrt 21)
[3 * 4] / 4.5 = AE = 8/3
+128633 Last one
Let the center of the largest semi-circle = E
Let the center of the next largest semi-circle = F
Let the center of the smallest semi-circle = D
Let the center of the circle = O.....let its radius = r
OE = 8 - r
OF = 5 + r
OD = 3 + r
DE = 5
FE = 3
cos (OEF) = -cos OED
Law of Cosines
OF^2 = FE^2 + OE^2 - 2 [ FE * OE]cos (OEF)
OD^2 = DE^2 +OE^2 - 2[ DE * OE]cos (OED)
(5 + r)^2 = 3^2 + (8- r)^2 + 2[3 * (8-r)]cos (OED)
(3 + r)^2 = 5^2 + (8-r)^2 - 2[5 *(8-r)] cos (OED)
cos (OED) = [ (5+ r)^2 - 9 - (8 -r)^2 ] / [6 (8-r)]
cos (OED) = [ (3 + r)^2 - 25 - (8-r)^2 ] / [-10 (8 -r)]
[(5 + r)^2 - 9 - (8-r)^2] / 6 = [(3 +r)^2 -25 - (8-r)^2 ] / (-10)
[ r^2 + 10r + 25 - 9 - r^2 + 16r -64 ] /6 = [ r^2 + 6r + 9 -25 - r^2 + 16r - 64 ] / -10
[ 26r - 48 ] / 6 = [ 22r - 80 ] / -10
-260r + 480 = 132r - 480
960 = 392r
r = 960 / 392 = 120 / 49
