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Math Help - Triangle

  1. #1
    Member Veronica1999's Avatar
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    Triangle

    In triangle ABC, angle C is a right angle and CB>CA. Point D is located on BC so that angle CAD is twice angle DAB.
    If AC/AD = 2/3, then CD/BD = m/n, where m and n are relatively prime positive integers. Find m+n

    All I have is the sketch.
    I'm working on it now but I think I'll really need some hints.
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  2. #2
    Member Veronica1999's Avatar
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    If CD is square root 5 isn't BD half of square root 5?

    I know the numbers are wrong because they have to be integers but I'm just asking about the rates.
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  3. #3
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    Quote Originally Posted by Veronica1999 View Post
    If CD is square root 5 isn't BD half of square root 5?
    Nooooo....
    Veronica, this is not an "easy" problem; and we don't know "where you're at"...
    OK, let's see:
    start with the smaller right triangle ACD: make AC = 2 , AD = 3 ; so CD = sqrt(5) (as you show!).
    now, can you calculate angle CAD?

    extend CD to CB, such that angle CAB = 1.5 * angle CAD (makes angle CAD twice angle BAD);
    now, can you calculate length of BC?
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  4. #4
    Member Veronica1999's Avatar
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    I hope you'll be patient with me because my math is very fragmented.
    I learn theorems through the problems so certain areas I'm strong but others I hardly know the basics. But I'm a very quick learner.

    First, I changed the rate to 2square root 5 and 3 square root 5 so I can at least work with one integer 5 for CD.
    I calculated the value for arccos (0.666666667) =48.1896
    This makes the angle CAB 72.2796.
    tan 72.2796 =3.12956
    BD + 5 / 2square root5 = 3.129
    BD=8.992888 which I changed BD=9
    and got CD/BD =5/9.


    Am I allowed to change the value of 8.992888 to 9?



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  5. #5
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    [QUOTE=Veronica1999;587931]Am I allowed to change the value of 8.992888 to 9[QUOTE]
    No need to: it is already 9!
    Need to use more precise angles: 48.1896851042214... * (3/2) = 72.2845276563321...

    Good work

    10/18, 15/27, 20/36 ...... also solutions, but not coprimes.
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  6. #6
    Member Veronica1999's Avatar
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    I learned a lot!
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