Definitions related to circles

1

• Definitions related to circles
• Center, radius, diameter, chord, major arc, minor
  arc, tangent, secant
• Angles inside circles
• Central angle, inscribed angle, cyclic
  quadrilateral
• Circumference and arc length
• C pd or C 2pr
• Arc length (Circumference)(Arc measure / 360)

2
Chord Properties

• C-54 Chord Central Angles Conjecture
• If two chords in a circle are congruent, then
  they determine two central angles that are
  congruent
• C-55 Chord Arcs Conjecture
• If two chords in a circle are congruent, then
  their intercepted arcs are congruent
• C-56 Perpendicular to a Chord Conjecture
• The perpendicular from the center of a circle to
  a chord is the bisector of the chord
• C-57 Chord Distance to Center Conjecture
• Two congruent chords in a circle are equidistant
  from the center of the circle
• C-58 Perpendicular Bisector of a Chord
  Conjecture
• The perpendicular bisector of a chord passes
  through the center of the circle

3
Tangent Properties

• C-59 Tangent Conjecture
• A tangent to a circle is perpendicular to the
  radius drawn to the point of tangency
• C-60 Tangent Segments Conjecture
• Tangent segments to a circle from a point
  outside the circle are congruent
• Internally and externally tangent circles

4
Arcs and Angles

• C-61 Inscribed Angle Conjecture
• The measure of an angle inscribed in a circle is
  half the measure of the intercepted arc
• C-62 Inscribed Angles Intercepting Arcs
  Conjecture
• Inscribed angles that intercept the same arc are
  congruent
• C-63 Angles Inscribed in a Semicircle Conjecture
• Angles inscribed in a semicircle are right
  angles
• C-64 Cyclic Quadrilateral Conjecture
• The opposite angles of a cyclic quadrilateral
  are supplementary
• C-65 Parallel Lines Intercepted Arcs Conjecture
• Parallel lines intercept congruent arcs on a
  circle

5
The Circumference/Diameter Ratio

• The circumference of a circle is the distance
  around the circle
• The ratio of the circumference of a circle to the
  diameter of the circle is a constant, p (pi),
  which is approximately 3.1416 or 22/7
• C-66 Circumference Conjecture
• If C is the circumference and d is the diameter
  of a circle, then there is a number p such that C
  pd.
• If d 2r where r is the radius, then C 2pr.

6
Around the World

• Circles (and spheres) are fairly common, so p
  appears in many real world problems
• Tangential velocity is a measure of the distance
  an object travels along a circular path in a
  given amount of time
• Measured in distance per unit time, e.g., meters
  per second
• Computed by dividing the circumference of the
  circle by the amount of time required for a
  complete revolution
• Depends on the distance from the center of the
  circle
• Angular velocity is a measure of the rate at
  which an object revolves around the center of a
  circle or the axis of a sphere
• Measured in degrees per unit time, e.g., degrees
  per second
• Computed by dividing 360 by the amount of time
  required for a complete revolution
• Independent of the distance from the center of
  the circle

7
Arc Length

• Arc measure is equal to the degree measure of the
  central angle that intercepts the arc
• Arc length is the distance between endpoints of
  the arc as measured along the circumference of
  the circle
• C-67 Arc Length Conjecture
• The length of an arc equals the circumference of
  the circle times the arc measure divided by 360

8
Proving Circle Conjectures

• The Inscribed Angle Conjecture is the key to many
  proofs in this chapter
• Its proof can be split into three special cases
• The center of the circle is on the inscribed
  angle
• The center of the circle is outside the inscribed
  angle
• The center of the circle is inside the inscribed
  angle
• Thinking backwards is helpful in developing these
  proofs
• Remember that once one conjecture has been
  proven, it can be used to prove other conjectures