Two circles with centre at $A$ and $B$ touch at $T$. $BD$ is the tangent at $D$ and $TC$ is a common tangent. $AT$ has length 3 and $BT$ has length 2. The length of $CD$ is:
Step-by-Step Solution
Key Concept: The chord and secant lengths are determined by the cosine of the angle between the circle and the line.
From $BC = \frac{2}{\cos \theta} = \frac{2}{4/5} = \frac{5}{2}$, and $CD = 4 - \frac{5}{2} = \frac{3}{2}$. The key is recognizing that $B$, $C$, and the circle's tangent points form specific geometric relationships based on the angle $\theta$.
Correct Answer: 2