MathematicsState true or false.AB and AC are two equal chords of a circle. Then the bisector of the angle BAC passes through the center of the circle.

State true or false.


AB and AC are two equal chords of a circle. Then the bisector of the angle BAC passes through the center of the circle.


  1. A
    True
  2. B
    False 

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    Solution:

    Given two equal chords of a circle.
    Let us assume AB and AC are two equal chords of a circle with center O and, AD intersect BC at point P.
    In triangle Δ  BAP and Δ  CAP,
    AB=AC
    ∠  BAP =∠  CAP
    AP=BP (common side)
    By SAS congruence rule, Δ  BAP ≅Δ  CAP
    And by CPCT theorem,
    ∠  BAP =∠  CPA
    Since perpendicular bisector AP bisect chord BC, then CP=CB.
    As ∠  BPA and  ∠  CPA are linear pair angles, so ∠  BPA+ ∠  CPA = 180 °  , then ∠  BPA =∠  CPA = 90 °  .
    Thus, the perpendicular bisector AP of chord BC is passing through the center O.
    Hence, the statement is true and option 1 is correct.
     
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