geometry

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given: chord AB is parallel to chord CD

prove if two chords of a circle are parallel, the two arcs between the chords are congruent.

Prove: arc AC is congruent to arc BD

  • geometry -

    Draw a figure by hand and insert all the points, lines as angles as directed below in order to follow the proof.

    Insert the centre of the circle and call it O, the radius of the circle is r.

    Draw another line parallel to AB and CD through O and call it EF, where E is between A and C, F is between B and D.

    Consider isosceles triangle ABO where AO=BO=r:
    α=∠OAB=∠OBA (isosceles triangle)
    Now consider parallel lines AB and EOF.
    ∠AOE=∠BAO (alternate angles)
    ∠BAO=∠ABO (isosceles triangle)
    ∠ABO=∠BOF (alternate angles)
    Therefore
    ∠EOA=∠FOB

    Similarly,
    ∠EOC=∠FOD

    Thus
    ∠AOC=∠BOD

    Length of arc AEC=r*∠AOC
    Length of arc BFD=r*∠BOD

    Therefore arc AEC = arc BFD

  • geometry -

    A circle has a diameter of 20 inches and a central angle AOB that measures 160°. What is the length of the intercepted arc AB? Use 3.14 for pi and round your answer to the nearest tenth

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