Saturday, August 13, 2016

Geometry Problem 1244: Circle, Radius, Perpendicular, Chord, Secant, Measurement. Mind Map

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

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Geometry Problem 1244: Circle, Radius, Perpendicular, Chord, Secant, Measurement. Mind Map. Mobile Apps
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5 comments:

  1. Peter TranAugust 13, 2016 at 10:26 PM

    https://goo.gl/photos/FGTGW9iSkgxRKfx16

    Connect BF, BC and BD
    Note that C is the midpoint of arc BG
    Triangle BEC similar to FBC ( case AA)
    So CB^2=CE.CF
    Triangle DCB similar to BCA ( case AA)
    So CB^2=CD.CA
    And CE.CF=CD.CA or 4x 10= 3(x+3)
    So x= 31/3

    ReplyDelete
  2. Sumith PeirisAugust 14, 2016 at 5:47 AM

    Let AB cut the circle at G.

    In isoceles Tr. BCG,

    CG^2 - CE^2 = BE.GE = CE.EF = 24
    So CG^2 = 24 + 16 = 40

    Now < CGB = CBG = < GDA

    Hence < CGD = < DGB - < CGB = < DGB - < GDA = < GAD

    Hence CG^2 = CD.CA

    So 40 = 3(3+x) from whence x = 31/3

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  3. Sumith PeirisAugust 14, 2016 at 10:47 AM

    2nd solution

    < CFD = < CGD = < EAD as before

    Hence AFED is cyclic and the result is easily calculated

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  4. UnknownJanuary 20, 2021 at 10:19 AM

    Join points F.O.D then WE CAN eaisly solve for r.
    Note that FO=OD=OC=r so Triangle FCD is a right triangle
    By similar angle WE have traingle FCD is similar to ECA so
    4/3+x=3/10 then X=31/3

    ReplyDelete
    Replies
    1. AnonymousJanuary 21, 2021 at 9:37 AM

      First you will need to prove the collinearity of F.O.D

      Delete

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