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Thread: proof: triangle of variable length on the Cartesian plane

  1. #1
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    proof: triangle of variable length on the Cartesian plane

    A variable triangle OAB is formed by a straight line passing through the point P(a, b) on the Cartesian plane and cutting through the x-axis and y-axis at A and B respectively.
    If the angle OAB=theta, find the area of triangle OAB in terms of a, b, and theta.
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  2. #2
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    Quote Originally Posted by shawli View Post
    A variable triangle OAB is formed by a straight line passing through the point P(a, b) on the Cartesian plane and cutting through the x-axis and y-axis at A and B respectively.
    If the angle OAB=theta, find the area of triangle OAB in terms of a, b, and theta.
    Hello Shawli:

    grad$\displaystyle =tan(\theta)$

    $\displaystyle tan(\theta)=\frac{b-B}{a} $

    $\displaystyle b-a.tan(\theta)=B$

    $\displaystyle tan(\theta)=\frac{b}{a-A}$

    $\displaystyle \frac{a.tan(\theta)-b}{tan(\theta)}=A$

    $\displaystyle Area=\frac{1}{2}.A.B$ (Substitute the values of A & B)

    Hope this helps
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    Name the points $\displaystyle B0,c)~\&~Ad,0)$.
    Then the area of the triangle is $\displaystyle \frac{cd}{2}$.
    The slope of the line is $\displaystyle -\tan(\theta)$.
    Use that to find $\displaystyle c~\&~d$.
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