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A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ 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.

 Jun 22, 2019
 #1
avatar+75 
-6

DOUBLE POST FOUND: https://web2.0calc.com/questions/a-few-theoretically-short-problems

 

A circle with center O has radius 8 units and circle P has radius 2 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.

 Jun 22, 2019
 #2
avatar+140 
+3

The links aren't working since the service was shut down.

 Jun 23, 2019
 #3
avatar+104962 
+4

I think this problem is solved easily if we lay it out in this manner :

 

 

 

 

Let the tangent to the circles be the equation y  = -8

Let the center of circle O   be  (0, 0)

Let the center of circle P  be (a, - 6)

 

Let the equations of the two circles be :

 

x^2 + y^2  = 64   

 

And

 

(x - a)^2  + (y - 6)^2  = 4

 

And ....since the distance between the centers of the circle is the sum of the radi, i,e.,  10 units, we can find the x coordinate of the center  of circle P, a,   as follows :

 

(a - 0)^2 + (0 - 6)^2  = 10^2

 

a^2  + 36  =  100

 

a^2  =  64

 

a  = 8

 

So...the center of  P  is (8, -6)

 

So  S  must  have  the coordinates (8, -8)

 

So....OS    has a length of     sqrt  [ (8 - 0)^2 + (-8 - 0)^2 ]  =  sqrt [ 64 + 64]  =  sqrt (64 * 2) =

 

8 sqrt(2) units

 

cool cool cool

 Jun 25, 2019
edited by CPhill  Jun 25, 2019

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