Math

• Circle

  • One of the most important shapes in geometry
  • Has 0 sides and is completely curved
  • Two-dimensional shape defined as a collection of the equidistant points on a plane from a certain fixed point
  • Measures 360°
• . Center

  • Equidistant from all of the ‘rim’ of the circle
  • Denotes the name of the circle
• Diameter

  • Chord that passes through the center of a circle
  • Cuts the circle into two equal parts
  • Line segment
• Radius

  • Half the diameter
  • Segment whose endpoints are the center and a point on the circle
• Chord

  • Doesn’t need to pass through the center
  • A segment whose endpoints are on the circle
  • All diameters are chords but not all chords are diameters
• Secant

  • Line that passes through the circle of two distinct points
  • Can be considered a chord
• Tangent

  • A line that touches one point in the circle
  • Point of tangency - point at which a line intersects a circle
• Central Angle

  • Formed by only two of the radii of a circle whose vertex is at the center
  • Intercepted arc - two other points
  • Whatever the measurement of the central angle is, it is equal to the intercepted arc
• Inscribed Angle

  • Angle whose vertex is on the circle and whose sides contain chords of the circle
  • If inscribed angle is 60°, intercepted arc is 120°
• • Central Angle-Intercepted Arc Postulate The measure of a central angle is equal to the degree of its intercepted arc
• Inscribed Angle Theorem

  • The measure of an inscribed angle is one-half the measure of its intercepted arc
• Arc Addition Postulate - The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs
• Semicircle Theorem - An angle inscribed in a semicircle is a right angle
• Quadrilateral inside a circle - If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary (180°)
• Secant - Latin word “secare” - to cut - A line that touches two points of a circle
• Tangent - Latin word “Tangere” - to touch - Touches only one point
• Two Secants-Interior Point Theorem

  • The measure of an angle formed by two secants intersecting in the interior is equal to half the sum of the measures of the arc intercepted
• Tangent-Secant on the Circle-Intercepted Arcs Theorem

  • The measure of angles formed by the intersection of a tangent and a secant on the circle is half the measure of its intercepted arc
  • Uses the same solution as inscribed angles
• Two Tangent-Exterior Point Theorem

  • Angle formed is equal to half the difference of the measure of the major and minor arc
• Tangent-Secant-Exterior Point Theorem

  • Half the difference of the measures of their intercepted arcs
• Two Secants-Exterior Point Theorem

  • Half the difference of the intercepted arcs
• Intersecting Chords Power Theorem Two chords intersect

  • Product of the lengths of the segment of one chord = product of the lengths of the segment of the other cord
  • Not necessarily center
• Intersecting Secants Power Theorem Intersects in the exterior of the circle

  • Lengths of one secant segment and its external part is equal to the product of the lengths of the other secant segment and its external part
• Tangent-Secant Power Theorem

  • Intersects at exterior
  • Square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part