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Circles $A$ and $B$ are externally tangent.  Angle $PAB$ is a right angle.  Segment $PT$ is tangent to circle $B$ at $T$.  If the radius of circle $A$ is $1$ cm and the radius of circle $B$ is $9$ cm, what is the length of segment $PT$ in cm?

 

 Mar 18, 2021
 #1
avatar+121006 
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Connect PB

 

Then triangle  PAB is  a right triangle   with legs  PA  and  AB  and hypotenuse  PB

 

PA  =  1             AB =  9  + 1 =   10 

 

PB  =   sqrt  ( AB^2  + PA^2)   =  sqrt  ( 10^2  + 1^2)  = sqrt (101)

 

Likewise angle PTB  is right  since  a radius  drawn to a tangent  meets  the  tangent at a  90°  angle

 

So triangle PTB is also right

 

So PB is  the  hypotenuse of  this triangle  and PT  and TB  are legs  with  TB  = 9

 

So

 

PT =  sqrt  (  PB^2  - TB^2)  =  sqrt  (101 - 9^2)  =  sqrt ( 101  - 81)  = 

sqrt (20)  =

sqrt(4) * sqrt (5)  =

2 sqrt (5) cm

 

 

cool cool cool

 Mar 18, 2021
edited by CPhill  Mar 18, 2021

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