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# Help with Geo

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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

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

Jun 23, 2019
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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

Jun 25, 2019
edited by CPhill  Jun 25, 2019