Monday, May 19, 2008

Elearn Geometry Problem 45



See complete Problem 45
Angles, Triangle. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

7 comments:

  1. Faltan datos, la repuesta asume que AC = CE.. y no se puede llegar a esa conclusion o en todo caso no explica como llega a ese punto.
    Espero respuestas.
    Muchas Gracias.

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  2. http://img842.imageshack.us/img842/3018/p045angletriangle.gif

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  3. Lets use a and b for alpha and beta. BD meets AC on E. Ang(ADE) = 2a+3b and ang(CDE) = a+b. Using sinus rule in ABD and BCD, we get
    AB/BD = sin (2a+3b) / sin (2a)
    and BC/BD = sin (a+b) / sin (a),
    so sin (2a+3b) / sin (2a) = sin (a+b) / sin (a).
    That equality gives
    sin (2a+3b) = 2.sin (a+b).cos (a)
    sin (2a+3b) = sin (2a+b) + sin (b)
    sin (2a+3b) - sin (2a+b) = sin (b)
    2.sin (b).cos (2a+2b) = sin (b)
    and cos (2a+2b)=1/2 or 2a+2b = 60º.
    Finally, x+2a = 90º-2b, x = 90º - (2a+2b) = 30º.

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  4. Typos corrected (regret inconvenience)
    Write a for alpha and b for beta.
    Draw BE the ⊥ bisector of AC, meeting AD at F.
    BE bisects ∠ABC(=4b)
    So ∠FBC = 2b,
    which implies BD bisects ∠FBC with ∠FBD = ∠DBC = b
    Next ∆FAC is isosceles.
    So ∠FCA = x, and
    ∠FCB = ∠FAB = 2a by symmetry
    which implies ∠FCD = ∠DCB = a
    Follows D is the incentre of ∆BFC, FD bisects ∠BFC.
    ∠BFD = ∠DFC = x + x = 2x
    Denote the sum a + b by y.
    From ∆BFC we have 4x + 2a + 2b = 180° and
    from ∆ABC we have 2x + 4a + 4b = 180°
    2x + y = 90° and x + 2y = 90°
    solving which we get x = y = 30°

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  5. http://s1092.photobucket.com/albums/i418/vprasad_nalluri/?action=view&current=elearngeometryproblem45.jpg

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  6. Figure for e-learn geometry problem 45 to follow my proof
    Link:
    http://imageshack.us/content_round.php?page=done&l=img198/5108/elearngeometryproblem45.png#

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  7. Denote alpha by @ and beta by £

    Let the bisector of < ABC meet AD at E. Now D is easily the incentre of Tr. BCE and so < BED = < CED = x.

    Since < BAC = 2@+x = < ACB, we have 2x + 4@ + 4£ = 180 .....(1)

    Since BE is perpendicular to AC,

    x + 2@ + 2£ = 90.....(2)

    From (1) and 2 we deduce that
    @ + £ = 30 and so from (1) x = 30

    Sumith Peiris
    Moratuwa
    Sri Lanka

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