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In triangle ABC, AC = 3, BC = 4, and AB = 5.  A semicircle with its center on segment AC is tangent to AB and BC.  Find the radius of the semicircle.

 Dec 21, 2019
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See the following image  :

 

 

 

Let the slope of AB  = -3/4

And the equation of the line  containing AB  is   y = (-3/4)x + 3

Put this line into standard form and we have that   

3x + 4y - 12   = 0 

 

And to find the distance  between  (0,0)  and this line we have that

 

l 3(0)  + 4(y)  - 12   l                12                     12

________________    =    _________   =       ___     =   2.4

   sqrt (3^2 + 4^2)                sqrt (25)                 5

 

Let  CE  be  perpendicular  to AB

 

And  let  the radius of  the  semicircle  = R

 

So....we  can construcst similar triangles   such that

 

  R                  2.4

_____   =     _____      cross-multiply

3 - R                 3

 

 

3R  = 2.4 ( 3 - R)

 

3R = 7.2 - 2.4R

 

5.4R  = 7.2

 

R =   7.2 / 5.4   =   4/3

 

 

cool cool cool

 Dec 21, 2019

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