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Math Help - Find Area of Triangle ABC

  1. #1
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    Find Area of Triangle ABC

    See picture.
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    MHF Contributor ebaines's Avatar
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    Re: Find Area of Triangle ABC

    An easy way to do this is to find the area of the large rectangle and subtract the areas of the three triangles that are not part of ABC.
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    Re: Find Area of Triangle ABC

    \displaystyle \begin{align*} \vec{AB} &= \left( 8, 5 \right) - \left( 3, 4 \right) \\ &= \left( 5, 1 \right) \\ \\ \vec{AC} &= \left( 7, 8 \right) - \left( 3, 4 \right) \\ &= \left( 4, 4 \right) \\ \\ \vec{AB} \times \vec{AC} &= \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 1 & 0 \\ 4 & 4 & 0 \end{matrix} \right| \\ &= \mathbf{i} \left| \begin{matrix}  1 & 0 \\ 4 & 0 \end{matrix} \right| - \mathbf{j} \left| \begin{matrix} 5 & 0 \\ 4 & 0 \end{matrix} \right| + \mathbf{k} \left| \begin{matrix} 5 & 1 \\ 4 & 4 \end{matrix} \right| \\ &= 0\mathbf{i} + 0\mathbf{j} + 16\mathbf{k} \\ \\ \left| \vec{AB} \times \vec{AC} \right| &= \sqrt{ 0^2 + 0^2 + 16^2} \\ &= \sqrt{16^2} \\ &= 16 \\ \\ \frac{1}{2} \left| \vec{AB} \times \vec{AC} \right| &= 8 \end{align*}

    So the area of the triangle is \displaystyle \begin{align*} 8\,\textrm{units}^2 \end{align*}
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    Re: Find Area of Triangle ABC

    Thank you everyone, especially Prove It. Great math work display.
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    Re: Find Area of Triangle ABC

    Find Area of Triangle ABC-19-nov-13-2.png
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