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Thread: angle trisector?

  1. #1
    Member Kaloda's Avatar
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    angle trisector?

    In triangle ABC, one of the trisector of angle A is a median, and the other is an altitude. If BC = 24, what is the area of triangle ABC?
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  2. #2
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    Quote Originally Posted by Kaloda View Post
    In triangle ABC, one of the trisector of angle A is a median, and the other is an altitude. If BC = 24, what is the area of triangle ABC?
    1. I've drawn a sketch (see attachment)

    2. $\displaystyle \Delta(AMC)$ is an isosceles triangle and therefore the height h bisects half of $\displaystyle \overline{BC}$.

    3. In the right triangle $\displaystyle \Delta(AMF)$ you know about the angle $\displaystyle \alpha$:

    $\displaystyle \tan(\alpha)=\dfrac6h$

    4. According to the question $\displaystyle \angle(BAM)=\alpha$.
    In the right triangle $\displaystyle \Delta(ABF)$ you know:

    $\displaystyle \tan(2\alpha)=\dfrac{18}h$

    5. Since $\displaystyle \tan(2\alpha)=\dfrac{2\tan(\alpha)}{1-(\tan(\alpha))^2}$ you'll get:

    $\displaystyle \dfrac{18}h = \dfrac{2\cdot \frac6h}{1-\left(\frac6h\right)^2}$

    Solve for h and subsequently calculate the area of the triangle $\displaystyle \Delta(ABC)$
    Spoiler:
    I've gotten: $\displaystyle Area(\Delta(ABC)) = 72\sqrt{3}$
    Attached Thumbnails Attached Thumbnails angle trisector?-winkl3teilung.png  
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  3. #3
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    I was looking over this, and am somewhat confused about step 5 the solution. I am not sure how you got the second half of that fraction. If possible, could you elaborate on that? Thanks!
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  4. #4
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    Quote Originally Posted by fireballs619 View Post
    I was looking over this, and am somewhat confused about step 5 the solution. I am not sure how you got the second half of that fraction. If possible, could you elaborate on that? Thanks!
    I'm not certain that I understand your question correctly ...

    1. This is a formula: $\displaystyle \tan(2\alpha)=\dfrac{2\tan(\alpha)}{1-\left(\tan(\alpha) \right)^2}$

    2. In step #3 you find $\displaystyle \tan(\alpha)=\dfrac6h$

    3. I then replaced $\displaystyle \tan(\alpha)$ in the formula by $\displaystyle \dfrac6h$

    ... or do you want me to derive the formula?
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  5. #5
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    Oh, my apologies, as I did not realize that it was a formula. I thought you had somehow derived it from the diagram, and I could not for the life of me figure out how. Thanks for the time, though!
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