Power of a Point (Easy)

AMC 10 Q1–8 depth — contest entry, not school drills.

Power of a Point: core idea

Intersecting chords.

Two chords intersect inside a circle: segments are $3$ and $8$ on one chord, $4$ and $x$ on the other. Find $x$ .

Hint: $3\cdot 8=4\cdot x$ . Answer: 6.

1. Intersecting chords: segments $2$ and $12$ ; on the other chord $3$ and $x$ . Find $x$ .
2. From external point, tangent length $6$ and secant external segment $4$ . Find whole secant length.
3. Chords intersect: $5\cdot x=4\cdot 10$ . Find $x$ .
4. Tangent $8$ , external secant segment $4$ . Whole secant?
5. Intersecting chords: segments $6$ and $6$ ; other chord $4$ and $x$ .
6. From external point, tangent length $6$ and whole secant length $9$ . Find the external secant segment.
7. Chords: $2$ and $18$ ; other side $6$ and $x$ .
8. Tangent $9$ , external segment $3$ . Whole secant?
9. Intersecting chords: $7$ and $4$ ; other chord $x$ and $2$ .