Let $\Delta ABC \sim \Delta DEF$ and their areas be,respectively,$64 \text{ cm}^2$ and $121 \text{ cm}^2$. If $EF = 15.4 \text{ cm}$,find $BC$ (in $\text{cm}$).

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

$ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.

In the figure,$\frac{QR}{QS} = \frac{QT}{PR}$ and $\angle 1 = \angle 2$. Show that $\Delta PQS \sim \Delta TQR$.

In the figure,$D$ is a point on side $BC$ of $\Delta ABC$ such that $\frac{BD}{CD} = \frac{AB}{AC}$. Prove that $AD$ is the bisector of $\angle BAC$.

Sides of a triangle are given below. Determine whether it is a right triangle. In case of a right triangle,write the length of its hypotenuse.
$3 \text{ cm}, 8 \text{ cm}, 6 \text{ cm}$