# Thread: Find Area of Triangle ABC

See picture.

2. ## 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.

3. ## Re: Find Area of Triangle ABC

\displaystyle \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 \displaystyle \begin{align*} 8\,\textrm{units}^2 \end{align*}

4. ## Re: Find Area of Triangle ABC

Thank you everyone, especially Prove It. Great math work display.