B. MATH- CIRCLES

• Circle: Set of points that are of the same distance from a given point in a plane
• Center: The given point
• Radius: The segment from the center to any point on the circle
• Chord: A segment whose endpoints both lie on the circle
• Diameter: A chord that passes through the center

  • Combination of 2 radii
• Interior of the circle: Set of points in the plane of a circle whose distances from the center are less than the length of the radius
• Exterior of the circle: Set of points in the plane of a circle whose distance from the center are greater than the length of the radius
• Theorems on the relationship between the radii and chords of circles:

  • Theorem 104: If a radius is perpendicular to a chord, then it bisects the chord
• Congruent circles: Circles that have congruent radii
• Concentric circle: Coplanar circles that have the same center
• Theorem 107: If chords of a circle or congruent circles are equidistant from the center(s), then the chords are congruent
• Theorem 108: If chords of a circle or of congruent circles are congruent, then they are equidistant from the centers of the circle
• Radius is found through the Pythagorean Theorem
• THEOREM 105

  • If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to the chord. 
• THEOREM 106

  • The perpendicular bisector of the chord passes through the center of the circle. 
• RADIUS IS FOUND THROUGH THE PYTHAGOREAN THEOREM
• Central angle: Angle whose vertex is the center of the circle
• Arc: Part of the circle that consists of 2 points on the circle and the unbroken part of the circle
• Endpoints of the arc: Two points on the circle
• 3 types of arc:

  • Minor arc
  • Major arc
  • Semicircle
• Minor arc:

  • Named using the 2 endpoints of the arc
  • 2 capital letters are used to name a minor arc
  • Less than one half of a circle
  • The minor arc is the union of 2 points and all points on the circle and in the interior of the central angle
• Major arc:

  • Named using 3 capital letters
  • More than one-half of a circle
  • Major arc ACB is the union of point a and b and all the points of the circle o in the exterior of central angle aob
• Semicircle:

  • Union of endpoints of a diameter and all points of the circle that lie on one side of the diameter
• The degree measure of a minor arc is equal to the degree measure of its central angle
• The degree measure of a major arc is equal to 360 minus the degree measure of its related minor arc
• The degree measure of a semicircle is 180
• Arc addition postulate: The measure of an arc formed by 2 adjacent, non-overlapping arcs is the sum of the measures of 2 arcs
• Congruent arcs: In the same circle or in congruent circles, arcs that have the same measure are congruent arcs
• Inscribed angle: Angle whose vertex lies on the circle and whose sides contain chords of the circle
• Theorem 115: The measure of an inscribed angle is equal to one-half measure of its intercepted arc
• 3 cases to consider:

  • Case 1: The center of the circle lies on one side of the inscribed angle
  • Case 2: The center of the circle lies on the interior of the inscribed angle
  • Case 3: The center of the circle lies on the exterior of the inscribed angle
• Tangent: A line on the plane of a circle that intersects it exactly at one point
• Point of tangency: Point of intersection
• Tangent circles: Circles that are tangent to the same line at the same point
• Internally tangent: If the centers of 2 circles are on the same side of the tangent line
• Externally tangent:

  • Centers of 2 circles are on opposite sides of the tangent line
• Common tangent:

  • A line tangent to 2 or more circles at different points
• Common external tangent:

  • Common tangent that does not intersect the segment joining the 2 centers of the circles
  • Circles on the same side
• Common internal tangent:

  • Common tangent that intersects the segment joining the 2 centers of the circles
  • Opposite sides
• Secant:

  • A line that intersects the circle at exactly 2 points