Circle properties
Check and apply conditions relating circle angles, chords, and tangents.
Meaning
Circles have fixed relationships among central angles, inscribed angles, chords, and tangents. An inscribed angle is half the central angle subtending the same arc; a tangent is perpendicular to the radius at contact. Joining the center to the inscribed-angle vertex and the arc endpoints forms isosceles triangles with equal radii. Their equal base angles and exterior-angle relationships give a central angle twice the inscribed angle. For a tangent, the shortest distance from the center to the line is the radius, so the contact radius is perpendicular to the tangent.
Example
A central angle of 100 degrees gives an inscribed angle of 50 degrees on the same arc. Their vertices are respectively at the center and on the circle. Two endpoints determine two arcs. Identify the intercepted arc; do not use the arc containing the inscribed angle's vertex.
Practice for this concept
Scope and references
Topics
Central and inscribed angles, chord and tangent properties, circles and lines
Learning goal
Check and apply conditions relating circle angles, chords, and tangents.
Helpful background
Check circle elements and triangle angle relationships.
Circle properties
These choice exercises use stated geometric conditions. They do not establish construction, drawing, spatial manipulation, written proof, or mastery of the whole geometry area.