Circle Geometry (Hard)

Late AMC 10/12 through early AIME — dense computation.

Circle Geometry: core idea

Cyclic quads.

A chord of length $16$ is $6$ from the center. Find radius.

Hint: Use the relevant chord, tangent, arc, or cyclic-angle theorem. Answer: 10.

1. Tangents $PA,PB$ have $PA=13$ and chord $AB=10$ . Find $PB$ .
2. An inscribed angle intercepts a $124^\circ$ arc. Find the angle.
3. A $72^\circ$ arc in radius- $10$ circle has what length?
4. Diameter $AB$ with $AC=9,BC=12$ . Find the circle radius.
5. Intersecting chords have pieces $3,12$ and $4,x$ . Find $x$ .
6. A tangent is $12$ ; a secant has external part $8$ and internal $x$ . Find $x$ .
7. A circle has circumference $20\pi$ . Find area of a $90^\circ$ sector.
8. Cyclic quadrilateral has $\angle A=73^\circ$ . Find opposite angle $C$ .
9. A radius- $13$ circle has chord $10$ . Find center-to-chord distance.