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Thread: orthocenter

  1. #1
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    Smile orthocenter

    i need to find the orthocenter of a triangle
    the cooridinates are:
    X = 55, 51
    Y= 10 , 4
    Z = 49, 95

    and if you can, can you show work


    Thank You very much for your help
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  2. #2
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    Orthocentre

    Hello misterriceman
    Quote Originally Posted by misterriceman View Post
    i need to find the orthocenter of a triangle
    the cooridinates are:
    X = 55, 51
    Y= 10 , 4
    Z = 49, 95

    and if you can, can you show work


    Thank You very much for your help
    Suppose that the orthocentre, O, has coordinates $\displaystyle (a, b)$. Then we use the fact that the line joining the orthocentre to any vertex is perpendicular to the opposite side. So:

    Gradient of OX = $\displaystyle \frac{b-51}{a-55}$

    Gradient of YZ = $\displaystyle \frac{91}{39}$

    So the product of these gradients is $\displaystyle -1$; i.e.

    $\displaystyle \frac{91}{39}\times\frac{b-51}{a-55}= -1$ (1)

    In a similar way, using OZ and XY, we get:

    $\displaystyle \frac{47}{45}\times\frac{b-95}{a-49} = -1$ (2)

    Now solve the simultaneous equations (1) and (2) to find $\displaystyle a$ and $\displaystyle b$. (Horrible numbers!) I make this:

    $\displaystyle a = \frac{382}{3}$ and $\displaystyle b = 20$

    I think you should check my working!

    Grandad
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  3. #3
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    thank u very muchhh
    i love u <3333333
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