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A circle with center O has radius 8 units and circle P has radius 3 units. The circles are externally tangent to each other at point Q. Segment TS is the common external tangent to circle O and circle P at points T and S, respectively. What is the length of segment OS? Express your answer in simplest radical form.

 Nov 24, 2020
 #1
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The length of segment OS ≈ 11.504 units

 Nov 24, 2020
 #2
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See the following  :

 

 

Let  the circle with a radius of 8  be centered at the origin

 

Let the circle  with  the radius of 3  be centered at  P  = ( a , -5) 

 

The centers of the circles will be 11 units apart

 

And we  can   use a right triangle  with a hypotenuse of  11 = OP  and one leg of 5 = RP .....to find a  = OR we have

 

sqry [ 11^2  - 5^2] =  sqrt [96]  =  6sqrt (4)  =  OR  = a 

 

O = (0,0)     S = ( 4√6 , -8 )  = ( √ 96, -8)

 

So    OS   =sqrt   [ (√96)^2  + (-8)^2  ] =   sqrt  [ 96 + 64 ]  =  sqrt [ 160 ]  =  4 sqrt (10)  

 

 

cool cool cool

 Nov 25, 2020
 #3
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Using Phill's diagram...

 

PR = SR - SP

 

OR = sqrt(OP2 - PR2)

 

OS = sqrt(OR2 + SR2)

 

 Nov 26, 2020

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