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?
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