Question

$overleftrightarrow{ap}$ is tangent to circle $o$ at $a$, circle $o$ has a radius of $6$ ft, circle $p$ has a radius of $2$ ft, and $ab = 6$ ft.
what is $cd$?
$\bigcirc$ $3$ ft
$\bigcirc$ $1$ ft
$\bigcirc$ $2$ ft
$\bigcirc$ $4$ ft

Explanation:

Step1: Use tangent-radius theorem

$\angle OAP = 90^\circ$, so $\triangle OAP$ is right-angled.

Step2: Calculate length of $OP$

$OA=6$ ft, $AP=AB+BP=6+2=8$ ft.
By Pythagoras:
$$OP=\sqrt{OA^2 + AP^2}=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ ft}$$

Step3: Compute length of $CD$

$OC=6$ ft, $PD=2$ ft.
$CD=OP - OC - PD=10-6-2=2$ ft

Answer:

2 ft