Plane Geometry (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · ACT Math

Plane Geometry: core idea

Composite area decomposition.

A circle has radius $13$ and a chord of length $10$ . Find the distance from center to chord.

Hint: Use half the chord with a radius. Answer: 12.

1. A triangle's sides are $7,24,25$ . Find the altitude to the side of length $24$ .
2. Two similar polygons have perimeter ratio $4:7$ . If smaller area is $48$ , find larger area.
3. An inscribed angle intercepts an arc of $130^\circ$ . Find the angle.
4. A regular decagon has side $5$ . Its perimeter is what?
5. A semicircle has diameter $16$ . Find its perimeter in terms of $\pi$ .
6. A square is inscribed in a circle of radius $5\sqrt2$ . Find the square's area.
7. A triangle has sides $9,10,17$ . Find its area.
8. A regular polygon has interior angle $156^\circ$ . How many sides does it have?
9. A cone's slant height is $13$ and radius is $5$ . Find its height.