Power of a Point (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Power of a Point: core idea

Tangent-secant.

Circle radius $13$ , point $5$ from center. Chord through that point — find chord length.

Hint: Half-chord $=\sqrt{13^2-5^2}=12$ . Answer: 24.

1. Power of point $P$ w.r.t. circle radius $5$ : $OP=13$ . Find power.
2. Tangent length $10$ , whole secant $25$ . External secant segment?
3. Two circles intersect at $A,B$ . Line through $P$ on $AB$ meets circles at $C,D$ with $PC=4$ , $PD=9$ . If $PA=3$ , find $PB$ .
4. Point $(3,4)$ w.r.t. circle $x^2+y^2=25$ . Find power of the point.
5. External secant: external part $5$ , whole secant $20$ . Tangent length?
6. Radius $10$ , chord length $12$ . Distance from center to chord?
7. Intersecting chords: $8$ and $3$ ; other chord $4$ and $x$ .
8. Tangent length $12$ , external secant segment $4$ . Find whole secant length.
9. Power of $(4,3)$ w.r.t. $x^2+y^2=9$ ?