Circle Geometry (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Circle Geometry: core idea

Inscribed angles.

A chord has length $16$ in a circle of radius $10$ . Find its distance from the center.

Hint: Draw the radii and use perpendicular/chord or angle theorems. Answer: 6.

1. Two radii form a $120^\circ$ sector of radius $6$ . Find the arc length in terms of $\pi$ .
2. Two chords intersect: segments $3,8$ on one and $4,x$ on the other. Find $x$ .
3. An exterior angle formed by two secants intercepts arcs $140^\circ$ and $60^\circ$ . Find the angle.
4. A tangent and radius meet at a point on the circle. Find their angle.
5. An inscribed angle is $35^\circ$ . Find its intercepted arc.
6. A circle has circumference $20\pi$ . Find its area in terms of $\pi$ .
7. Find the length of a $60^\circ$ arc in a circle of radius $12$ in terms of $\pi$ .
8. Two tangent segments from one external point have lengths $x+2$ and $11$ . Find $x$ .
9. A chord $10$ units long is $12$ from the center. Find the radius.