Angles and segments in a circle
Inscribed and central angles, the angle on a diameter, tangents, cyclic quadrilaterals, intersecting chords, arc length and the area of a sector.
How to solve it
The main rule of the circle: an inscribed angle is half the central angle on the same arc. Everything else follows: an angle on a diameter is a right angle, and the opposite angles of a cyclic quadrilateral add up to $180^\circ$.
Step by step
- Find the arc the angle stands on. An inscribed angle is half of that arc; a central angle equals it.
- An angle standing on a diameter is $90^\circ$ — then the angle sum of the triangle does the rest.
- A tangent is perpendicular to the radius at the point of contact; the two tangent segments from one point are equal. The tangent length comes from Pythagoras: $PT^2 = PO^2 - r^2$.
- Chords crossing at a point $P$: $AP \cdot PB = CP \cdot PD$.
- An arc and a sector are a share of the whole circle: the angle over $360^\circ$.
Common mistakes
- Taking the wrong arc: an inscribed angle stands on the arc that does not contain its vertex. If the point is on the minor arc, the angle stands on the major one.
- Treating an inscribed angle as equal to the central one. It is half.
- Forgetting in tangent problems that the radius meets the tangent at a right angle: the quadrilateral $PT_1OT_2$ has two $90^\circ$ angles.
- Using the circumference instead of the area for a sector: $S = \pi r^2 \cdot \frac{\alpha}{360^\circ}$.