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Geometry

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Can you solve these problems?
1.

2.

3.

4.

Mar 17, 2024

#1
+129690
+1

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)

Mar 17, 2024
#2
+129690
+1

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

Mar 18, 2024
#3
+129690
+1

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

Mar 18, 2024
edited by CPhill  Mar 18, 2024
#4
+85
0

You are correct!

Mar 18, 2024