In $\Delta XYZ$, length of $XY = 13$ cm. If the length of the median drawn from vertex $Z$ to base $XY$ is $6.5$ cm, then $\Delta XYZ$ is a/an:

Let the median from $Z$ meet $XY$ at $M$. A median meets the opposite side at its midpoint, so:

$XM = MY = \dfrac{13}{2} = 6.5$ cm

$ZM = 6.5$ cm (given)

So $M$ is at the same distance ($6.5$ cm) from $X$, $Y$ and $Z$.

A circle with center $M$ and radius $6.5$ cm passes through $X$, $Y$ and $Z$, and $XY$ is its diameter.

An angle made on a circle by a diameter (angle in a semicircle) is always $90^\circ$.

So $\angle XZY = 90^\circ$.

Therefore, $\Delta XYZ$ is a right angled triangle.