Circle Theorems: Chords

Secant and Tangent Lines

  • Secant Line: A line that intersects a circle at two points.
  • Tangent Line: A line that intersects a circle at exactly one point.

Chords

  • Definition: A chord is a line segment connecting two points on a circle.
  • If points A and B lie on a circle, segment AB is a chord.
  • Diameter: A chord that passes through the center of the circle.
  • Most chords are not diameters.

Chord Properties

  • A chord (segment AB) and its corresponding arc (arc AB) are different. The chord is a straight line, while the arc follows the curvature of the circle.
  • Every chord has an arc associated with it (minor and major arcs).
  • A diameter has two semicircles associated with it.

Theorems

Theorem 1

  • Statement: Within a circle or congruent circles (circles with the same radius), two central angles are congruent if and only if their corresponding chords and arcs are congruent.
  • Congruent Circles: Circles with the same radius.
  • If two chords in a circle (or congruent circles) are congruent, then their corresponding central angles are congruent.
  • If two central angles in a circle (or congruent circles) are congruent, then their corresponding chords are congruent.
  • If ∠A≅∠B\angle A \cong \angle B, then chord AB is congruent to chord CD (and vice versa).
  • Since the central angles are the same, that means the arcs are the same as well.
Proof Outline
  1. Given: Congruent chords.
  2. Central Angles: Draw central angles corresponding to the chords.
  3. Isosceles Triangles: Recognize that radii create isosceles triangles.
  4. Side-Side-Side (SSS): Prove triangle congruence using SSS (all radii are equal, and chords are congruent by assumption).
  5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC): Conclude that central angles are congruent.
  • For the converse (congruent central angles imply congruent chords), use Side-Angle-Side (SAS) to prove triangle congruence.

Theorem 2

  • Statement: Within a circle or congruent circles, two chords are congruent if and only if their corresponding arcs are congruent.
  • If chord AB is congruent to chord CD, then arc AB is congruent to arc CD (and vice versa).
  • If arcs have the same measure and the same radius, they have to be congruent arcs

Theorem 3

  • Statement: Within a circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.
  • "Equidistant" refers to the perpendicular distance from the center to the chord.
  • If chord AB is congruent to chord CD, then the perpendicular distance from the center to AB is equal to the perpendicular distance from the center to CD (and vice versa).
Proof Outline
  1. Given: Congruent chords.
  2. Draw Perpendiculars: Draw segments from the center perpendicular to each chord.
  3. Radii: Draw radii to the endpoints of the chords.
  4. Hypotenuse-Leg (HL): Use the Hypotenuse-Leg theorem to prove the congruence of the right triangles formed (the hypotenuse is the radius, and the leg is half the chord length).
  5. CPCTC: Conclude that the perpendicular distances are congruent.
  • For the converse (equidistant chords imply congruent chords), use Side-Side-Side (SSS) to prove that the triangles are congruent.

Theorem 4

  • Statement: In a circle, a diameter is perpendicular to a chord if and only if it bisects the chord and its arc.
  • If diameter is perpendicular to chord AB, then chord AB is bisected, and arc AB is bisected (and vice versa).
Proof Outline
  1. Given: Diameter perpendicular to chord.
  2. Radii: Draw radii to the endpoints of the chord.
  3. Hypotenuse-Leg (HL): Use the Hypotenuse-Leg theorem to prove the congruence of the right triangles formed.
  4. CPCTC: Conclude that the chord is bisected.

Theorem 5

  • Statement: In a circle, the perpendicular bisector of a chord contains the center of the circle.
  • If you construct the perpendicular bisector of a chord, that line will pass through the center of the circle.
  • Any point on the perpendicular bisector is equidistant from the endpoints of the segment. Since the center of the circle is equidistant from the endpoints of the chord (both are radii), the center must lie on the perpendicular bisector.

Application of Theorem 5

  • Finding the Center of a Circle Using Only a Portion of the Circle:

    1. Draw a chord on the circle.
    2. Construct the perpendicular bisector of that chord.
    3. Draw another chord on the circle.
    4. Construct the perpendicular bisector of the second chord.
    5. The intersection of the two perpendicular bisectors is the center of the circle.