Given: Circle k(O) with OT ⊥ XY, and OU ⊥ WZ and X=Z. Prove: XY=WZ

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Given: Circle k(O) with OT ⊥ XY, and OU ⊥ WZ and X=Z. Prove: XY=WZ

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Yến Oanh 7 months 2021-08-09T19:51:03+00:00 1 Answers 2 views 0

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    2021-08-09T19:52:33+00:00

    ΔXOY ≅ ΔZOW ⇒ proved down
    Step-by-step explanation:
    * Lets study some facts on the circle
    – If two chords equidistant from the center of the circles,
    then they are equal in length
    * the meaning of equidistant is the perpendicular distances
    from the center of the circle to the chords are equal in length
    * Lets check this fact in our problem
    ∵ XY and WZ are two chords in circle O
    ∵ OT ⊥ XY
    – OT is the perpendicular distance from the center to the chord XY
    ∵ OU ⊥ WZ
    – OU is the perpendicular distance from the center to the chord WZ
    ∵ OT ≅ OU
    – The two chords equidistant from the center of the circle
    ∴ The two chords are equal in length
    ∴ XY ≅ WZ
    * Now in the two triangles XOY and ZOW , to prove that
    they are congruent we must find one of these cases:
    1- SSS ⇒ the 3 sides of the 1st triangle equal the corresponding
    sides in the 2nd triangle
    2- SAS ⇒ the two sides and the including angle between them
    in the 1st triangle equal to the corresponding sides and
    including angle in the 2nd triangle
    3- AAS ⇒ the two angles and one side in the 1st triangle equal the
    corresponding angles and side in the 2nd triangle
    * Lets check we will use which case
    – In the two triangles XOY and ZOW
    ∵ XY = ZW ⇒ proved
    ∵ OX = OZ ⇒ radii
    ∵ OY = OW ⇒ radii
    * This is the first case SSS
    ∴ ΔXOY ≅ ΔZOW

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