The lengths of the sides of a triangle are $5$ cm, $12$ cm and $13$ cm. Find the length of perpendicular from the opposite vertex to the side whose length is $13$ cm.

The correct option is B. $\frac{60}{13}$ cm

Let's say that the triangle is $ABC$ with $AB=12,BC=5,CA=13$ and $BD\perp AC$

As $AB$ and $BD$ are both perpendiculars( height ) of the $△ABC$ on the bases $BC$ and $AD$, respectively,

$\therefore ar\left(ABC\right)=\frac{1}{2}AB\times BC=\frac{1}{2}CA\times BD$.

$\Rightarrow BD=\frac{AB\times BC}{CA}=\frac{12\times 5}{13}=\frac{60}{13}$ cm.