Posers and Puzzles

Posers and Puzzles

  1. Joined
    21 Apr '05
    Moves
    54
    16 Jun '05 13:40
    In a triangle ABC, D and F are points on BC, and E is a point on AB such that AD is parallel to EF, AngleCEF = AngleACB, AD = 15m, EF = 8m and BF = 8m. Find the length of BC.
  2. Standard memberShanshu311
    Vox Populi
    38.9265 N, 94.6372 W
    Joined
    08 Sep '04
    Moves
    50349
    16 Jun '05 15:36
    You'd better tell us the answer, if nobody gets it. My geometry skills are very rusty, and the most I've gotten so far, is the length of BE.

    I suck.
  3. Mt. Doom
    Joined
    12 Apr '05
    Moves
    1524
    16 Jun '05 18:11
    OW! My head hurts!!!🙁😕
  4. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
    Moves
    52874
    16 Jun '05 18:51
    Originally posted by elopawn
    In a triangle ABC, D and F are points on BC, and E is a point on AB such that AD is parallel to EF, AngleCEF = AngleACB, AD = 15m, EF = 8m and BF = 8m. Find the length of BC.
    I may be all wet here but when I draw it out it looks like AD can
    be Perpendicular to EF but I don't see any way it could ever be
    parallel. Those two lines cross inside the ABC triangle and doesn't
    look to me like crossing lines can ever be parallel at least not in
    OUR universe.
  5. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
    Moves
    52874
    16 Jun '05 19:17
    Originally posted by elopawn
    In a triangle ABC, D and F are points on BC, and E is a point on AB such that AD is parallel to EF, AngleCEF = AngleACB, AD = 15m, EF = 8m and BF = 8m. Find the length of BC.
    looking at it again, could you have made a typo, meaning E is on
    a point AC instead of AB? that could allow the two lines to be
    parallel but they would still be called AD and FE.
  6. Standard memberShanshu311
    Vox Populi
    38.9265 N, 94.6372 W
    Joined
    08 Sep '04
    Moves
    50349
    16 Jun '05 19:38
    Originally posted by sonhouse
    I may be all wet here but when I draw it out it looks like AD can
    be Perpendicular to EF but I don't see any way it could ever be
    parallel. Those two lines cross inside the ABC triangle and doesn't
    look to me like crossing lines can ever be parallel at least not in
    OUR universe.
    Are you sure? I was able to draw it out, just fine.

  7. Joined
    07 Jul '04
    Moves
    102838
    16 Jun '05 19:51
    Hi sonhouse!

    just switch your points B and C and make sure D is closer to C than F is and then you can draw it. AB is the base of the triangle and C is the apex. So, going clockwise around the triangle starting at the lower left corner you would have A, C, D, F, B and E.

  8. Standard memberThe Plumber
    Leak-Proof
    under the sink
    Joined
    08 Aug '04
    Moves
    12493
    16 Jun '05 19:56
    Originally posted by sonhouse
    I may be all wet here but when I draw it out it looks like AD can
    be Perpendicular to EF but I don't see any way it could ever be
    parallel. Those two lines cross inside the ABC triangle and doesn't
    look to me like crossing lines can ever be parallel at least not in
    OUR universe.
    Construct an isoceles triangle where the two equal length sides are each 8m long - have the third side be the bottom of the triangle (to make this easier). Label the bottom two vertices B & E, and the top vertice F. Draw a line parallel to EF and 15m long such that its two ends will be along the extended lines BF and BE. Label the ends of these two lines A (bottom) and D (top). Then extend BD out some as yet to be determined distance and where it ends will be point C. Connect C to A, and you have the problem as described above. Note that BE and BF are both 8m long, and BD and AD are both 15 m long. Now the hard part is when you draw the line from C to E, the angle ACB has to equal the angle CEF.

    I;m with Shanshu, my geometry is just a little to rusty to finalize the solution (although I'm wondering how he calculated the length of BE....)
  9. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
    Moves
    52874
    16 Jun '05 21:08
    Originally posted by jebry
    Hi sonhouse!

    just switch your points B and C and make sure D is closer to C than F is and then you can draw it. AB is the base of the triangle and C is the apex. So, going clockwise around the triangle starting at the lower left corner you would have A, C, D, F, B and E.

    Isn't that interesting! I just automatically drew it ABC, an
    equalateral triangle with A in lower left hand corner, B on top and
    C on right hand tip. They are marked going clockwise.
    But if you draw it where they are going counterclockwise it puts
    EF and AD in parallel like the original description. Learn something
    new every day. Didn't know that would make a differance.
    Haven't even started solving the problem as of yet, just didn't get
    past step 1🙂
  10. Standard memberShanshu311
    Vox Populi
    38.9265 N, 94.6372 W
    Joined
    08 Sep '04
    Moves
    50349
    16 Jun '05 21:25
    I;m with Shanshu, my geometry is just a little to rusty to finalize the solution (although I'm wondering how he calculated the length of BE....)[/b]
    That would be the Pythagorean Theorum.

    😉
  11. Joined
    30 Oct '04
    Moves
    7797
    16 Jun '05 22:202 edits
    hmm... parallel lines& an angle
    Using Thales Theorem:
    EF/AD=BF/BD=BE/AB = 8/15; Hence BD=15;
    All that remains is CD=?
    But I can't seem to plug in the angle in the equation 🙁
    EDIT: One more thing though SIMILAR TRIANGLES
    Namely if we denote the intersection of CE and AD with H, we can easily see that the angle CHD = CEF (since AD||EF), and from the condition we get
    the angle ACB = CHD, and therefore the triangle ADC and CHD are similar.
    What good can be derived from that is a question I am currently unable to answer.
  12. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
    Moves
    52874
    16 Jun '05 22:30
    Originally posted by elopawn
    In a triangle ABC, D and F are points on BC, and E is a point on AB such that AD is parallel to EF, AngleCEF = AngleACB, AD = 15m, EF = 8m and BF = 8m. Find the length of BC.
    In this problem you can prove AD=DB, 15 units, so DF=7 units
    (FB is 8 so 15 minus 8 =7)
    So this proves only that the triangle ADB is isosceles.
    So building ADB says nothing about ABC, since there are an infinite
    number of possible lengths of AC that will satisfy the ADB requirement.
  13. Joined
    30 Oct '04
    Moves
    7797
    16 Jun '05 22:55
    Originally posted by ilywrin
    hmm... parallel lines& an angle
    Using Thales Theorem:
    EF/AD=BF/BD=BE/AB = 8/15; Hence BD=15;
    All that remains is CD=?
    But I can't seem to plug in the angle in the equation 🙁
    EDIT: One more thing though SIMILAR TRIANGLES
    Namely if we denote the intersection of CE and AD with H, we can easily see that the angle CHD = CEF (since AD||EF), and from the co ...[text shortened]... are similar.
    What good can be derived from that is a question I am currently unable to answer.
    And of course (stupid me) The Similar Triangles are three:
    ADC, CHD, and EFC 🙂
    Okay ratios:
    CD/AD=HD/CD (unknown CD and HD 🙁 )
    CF/AD=EF/CD may be rewritten as 7/15+CD/15 = 8/CD,
    Reworking that we get (CD = x for convenience)
    7x+x^2=120 or x^2 +7x - 120 = 0;
    Therefore the psoitive root, which is luckily 8 is the solution.
    Thus BC = 8+15 = 23
  14. Joined
    29 Apr '05
    Moves
    827
    16 Jun '05 22:57
    I agree with sonhouse there. But maybe we have to pay attention to the angles too, nobody did that so far. So my question is...the angle ACB means if you only look at that triangle, take the angle at the point C right? and for angle CEF you take that triangle CEF only and then take the angle at the point E?
  15. Joined
    30 Oct '04
    Moves
    7797
    16 Jun '05 23:21
    Well the only thing the equality of the angles may be used for is finding suitable triangles and then using the ratio to express the length of CD.
    Therefore, as my previous post claims, under the given conditions, BC must equal 23m.
Back to Top