Properties of Circles

Circles

Definition of a Circle

  • A circle is the set of all points equidistant from a fixed point (the center).
  • The distance from the center to any point on the circle is called the radius.
  • All radii of the same circle are congruent.
  • Interior of the circle: Points inside the circle.
  • Exterior of the circle: Points outside the circle.

Lines Intersecting Circles

  • A line can intersect a circle at two points, one point, or no points.
Secant Line
  • A secant line intersects a circle at two points.
Tangent Line
  • A tangent line is a line in the plane of the circle that intersects the circle at exactly one point.
  • The line must be in the same plane as the circle.
  • The point where the tangent line intersects the circle is called the point of tangency.
  • A radius drawn to the point of tangency is perpendicular to the tangent line.
  • If a line is tangent to a circle, then the radius drawn to the point of tangency is perpendicular to the line, and vice versa (if and only if).

Proof Involving Tangent Lines

  • Given: Circle with center C, point P outside the circle, tangent lines PA and PB from point P to the circle (where A and B are the points of tangency).
  • Prove: Segment PB is congruent to segment PA.
Proof
  1. Draw radii CA and CB.
  2. CA is perpendicular to PA, and CB is perpendicular to PB (radius drawn to the tangent line).
  3. Draw line PC.
  4. Triangles PCA and PCB are right triangles.
  5. BC is congruent to AC because they are both radii of the circle.
  6. PC is congruent to PC (reflexive property).
  7. Triangle PCB is congruent to triangle PCA by the Hypotenuse-Leg Theorem.
  8. PB is congruent to PA because they are corresponding parts of congruent triangles (CPCTC).
Hypotenuse-Leg Theorem
  • If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

Theorems

  • Theorem 1: If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
  • Theorem 2: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
  • Theorem 3: If two tangent segments to a circle share a common endpoint outside the circle, then the segments are congruent.