Sunday, March 8, 2015

Geometry Problem 1094: Tangent Circles, Tangent Chord, Radius, Sagitta of Arc

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

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Online Math: Geometry Problem 1094: Tangent Circles, Tangent Chord, Radius, Sagitta of Arc.

4 comments:

  1. Observe that CDT are collinear,
    thus ab=CD×DT.

    Now since
    CD=c/cos∠MDT
    DT=2r cos∠QDT

    Hence, ab=2CR.

    ReplyDelete
  2. A typo I think
    It should be CD=c/cos∠MCD
    (ab = 2cr of course)

    ReplyDelete
    Replies
    1. Thanks Pravin. It's MCD, and ab=2cr (lower case).

      Delete
  3. Geometry solution

    O,Q,T are collinear

    OQ^2 = MD^2 + (QD-OM)^2

    So { (a-b)/2}^2 + (r-R+c)^2 = (R-r)^2.......(1)

    Applying Pythagoras to Tr. AOM

    (R-c)^2 + {(a+b)/2}^2 = R^2...,,,(2)

    (2) -(1) and simplifying using the difference of 2 squares

    ab + r(2R-2c-r) = 2rR -r^2

    Which further simplifies to

    ab = 2cr

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete