In the diagram, $\triangle A B C$ has an area of 84 and $A C=12$. Point $D$ is on $A C$ so that $B D$ is perpendicular to $A C$. What is the length of $B D$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_486fbaaa1172337e46fbg-1.jpg?height=393&width=309&top_left_y=1300&top_left_x=968)
The area of $\triangle A B C$ is equal to $\frac{1}{2} \times A C \times B D$.

It is given that the area is 84 and that $A C=12$.

Substituting, this leads to the equation $84=\frac{1}{2} \times 12 \times B D$ or $84=6 B D$, so $B D=14$.

ANSWER: 14