In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $\frac{MB}{MD} = \frac{5}{3}$ and $CD = 12$,find $AB$.

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

In $\Delta PQR, m\angle Q = 90^{\circ}$ and $\overline{QM}$ is a median. If $PQ = 20$ and $QR = 21$,find $QM$.

In $\Delta ABC$,$\Delta PQR$,and $\Delta XYZ$,the correspondences $ABC \leftrightarrow QPR$ and $PQR \leftrightarrow YZX$ are similarities. Then,the correspondence $ABC \leftrightarrow \dots$ is a similarity.

In the given figure,if $AB \parallel DC$ and $AC$ and $PQ$ intersect each other at the point $O$,prove that $OA \cdot CQ = OC \cdot AP$.

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