Chapter 9

• What needs to happen in order for a polygon and a circle to be circumscribed/inscribed?

  Each vertex of the polygon must lie on the circle

  
• Concentric spheres

  Spheres that have the same center
• Concentric circles

  Circles that lie in the same plane and have the same center
• Sphere

  The set of all points in space a given distance from a given point called the center

  
• Congruent spheres

  Spheres that have congruent radii
• Congruent circles

  Circles that have congruent radii
• Secant

  A line that contains a chord

  
• Point of Tangency 


  The point at which the tangent intersects the circle

  
• Tangent

  A line in the plane of a circle that intersects the circle in exactly one point

  
• Diameter

  A chord that intersects the center of a circle

  
• Chord

  Segment whose endpoints lie on the circle

  
• Radius

  Distance from the center of the circle

  
• Circle

  The set of all points in a plane a given distance from a given point in that plane called the center

  
• If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.

  
• If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

  
• Tangents to a circle from the same point are congruent.

  
• Inscribed Circle

  When each side of a polygon is tangent to a circle, the circle is inscribed in the polygon.

  
• Tangent Circles

  Coplanar circles that are tangent to the same line at the same point

  
• Common EXTERNAL tangent

  Does not intersect the segment joining the centers

  
• Common INTERNAL tangent

  Intersects the segment joining the centers

  
• INTERNALLY tangent circles

  Circles that share common interior points

  
• EXTERNALLY tangent circles

  Circles that share no interior points

  
• Central Angle

  Angle with its vertex at the center of a circle

  
• Arc

  Unbroken part of a circle

  
• Minor Arc

  Arc of a circle that measures 0 < x < 180

  
• Major Arc

  Arc of a circle that measures 180 < x < 360

  
• Semicircle

  Arc of a circle that measures 180; endpoints are also endpoints of a diameter

  
• How do you find the measure of a minor arc?

  Equal to the measure of the central angle

  
• How do you find the measure of a major arc?

  360 - measure of its central angle

  
• How do you find the measure of a semicircle?

  Always 180
• Adjacent Arcs

  2 arcs in the same circle that share exactly one point

  
• Arc Addition Postulate

  The measure of the arc formed by adjacent arcs is the sum of the measures of these 2 arcs

  
• Congruent Arcs

  Arcs in the same circle or congruent circles that have the same measure

  
• In the same circle or 2 congruent circles, 2 minor arcs are congruent iff their central angles are congruent

  
• In the same circle or congruent circles

  1. Congruent arcs have congruent chords
  2. Congruent chords have congruent arcs
• In the same circle or congruent circles

  1. Chords equidistant from the center are congruent
  2. Congruent chords are equidistant from the center
• A diameter (or radius) that is perpendicular to a chord bisects both the chord and its arc
• Inscribed Angle

  An angle whose vertex is on a circle and whose sides contain chords of the circle

  
• The measure of an inscribed angle is equal to half the measure of its intercepted arc.

  
• If 2 inscribed angles intercept the same arc, then the angles are congruent.