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In a circle, two chords of lengths 4 and 11 respectively subtend central angles whose degree-measures are in the ratio 1 to 3 respectively.  Determine the length of the radius of the circle.

Dec 18, 2019

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By the Law of Cosines, we have that

4^2  =  2R^2  - 2R^2 ( cos A)

16 -2R^2

_______   =  (cos A)

-2R^2

R^2 - 8

_______  =   (cos A)

R^2

11^2  = 2R^2  - 2R^2 (cos 3A)

121 - 2R^2

__________   =   cos(3A)

- 2R^2

2R^2  - 121

_________  =  cos(3A)

2R^2

Using the identity   (cos3A)  =    4 (cos A)^3 - 3cosA

2R^2  - 121               4  [ R^2 - 8 ] ^3            3  [ R^2 - 8 ]

__________   =        _____________   -    ____________

2R^2                          R^6                           R^2

2R^2 - 121        6 [ R^2 - 8]                   4 [ R^2 - 8]^3

_________  + ___________   =           ____________

2R^2             2R^2                                    R^6

8R^2 - 169                 4 [ R2 - 8 ] ^3

__________   =         ____________

2R^2                          R^6

R^4 [8R^2  - 169]  =  8 [ R^2 - 8 ] ^3

8R^6 - 169R^4  =  8  [ R^6  - 24R^4 + 192R^2  - 512 ]

8R^6  - 169R^4  =  8R^6 -  192R^4 + 1536R^2 - 4096

23R^4  - 1536R^2  + 4096  =  0

Factor  as

(23R^2  -  64)  (R^2 - 64)  =  0

So  either

23R^2  - 64  = 0

R^2  =  64 / 23

R = ± √[64/ 23]   ≈  ±1.66 units  ( reject.....the diameter would be shorter than either chord....which is impossible )

Or

R^2  -  64   =  0

R  =   8  units = the radius

Dec 18, 2019