Plane Geometry
Plane Geometry tests triangle and circle properties, angles, lengths, areas, similarity, the power of a point, and inscribed figures. Diagrams are usually described in text, so translate each condition into an equation.
Core rules
- Inscribed angle theorem: an angle subtended by an arc at the circumference is half the central angle; angles in the same segment are equal.
- A tangent is perpendicular to the radius at the point of contact. Tangent-chord angle equals the angle in the alternate segment.
- Similar figures have equal corresponding angles and proportional corresponding sides; areas scale by the square of the similarity ratio.
- Power of a point: for an external point P, PT^2=PA·PB for tangent PT and secant PAB.
- The area of a triangle can be found from two sides and the included angle using 1/2 ab sin C.
Worked example
From external point P, tangent PT has length 6. A secant through P meets the circle at A then B, with PA=4. Find PB. By power of a point, PT^2=PA·PB, so 36=4·PB and PB=9. Then AB=PB-PA=5.
Traps
Match corresponding vertices in similarity statements; writing AB/DE=BC/EF is safe only with the correct correspondence. Do not use the tangent length as a secant segment. In cyclic quadrilaterals, opposite angles sum to 180°, not 90°. Check whether a length is a whole secant or only the external part.