In the given figure,$\angle BAC = 90^{\circ}$ and $AD \perp BC$. Then,

Diagonals of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at the point $O$. If $AB = 2 CD$,find the ratio of the areas of triangles $AOB$ and $COD$.

In the figure,$ABD$ is a triangle right-angled at $A$ and $AC \perp BD$. Show that $AB^2 = BC \cdot BD$.

It is given that $\triangle ABC \sim \triangle PQR,$ with $\frac{BC}{QR} = \frac{1}{3}.$ Then,$\frac{\operatorname{ar}(\triangle PRQ)}{\operatorname{ar}(\triangle BCA)}$ is equal to

If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB = 5, BC = 7, AC = 10$ and $PR = 15$,the perimeter of $\Delta PQR$ is........

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QD}$ is an altitude to the hypotenuse $\overline{PR}$. If $PQ = 4QR$,prove that $PD = 16RD$.