Chapter 9: Tangents and Secants to a Circle

Tangents and Secants to a Circle establishes that a tangent is always perpendicular to the radius at the point of contact, that two tangents from an external point are always equal in length, and covers constructing tangents and working with a secant's segments.

Forgetting to mark the tangent-radius right angle is what makes many problems here look like they're missing information, when they aren't — it's usually the key that unlocks the rest, turning an otherwise unsolvable diagram into a straightforward right-triangle problem. This chapter extends Class 9's Circles into a new context, reusing the same congruence and angle-chasing techniques built up across two years.