If one diagonal of trapezium divides the other diagonal in the ratio $1:2$. Prove that one of the parallel sides is double the other.

From the given figure, If$\angle MLN=k°$ and $△ABC~△MLN$, then find the value of $k$.

In a right angled triangle if hypotenuse is $20 \mathrm{cm}$ and the ratio of the other two sides is $4:3$, find the sides.

In the given figure, $DE\parallel AC$ and $DF\parallel AE$. Prove that $\frac{EF}{BF}=\frac{EC}{BE}$.

In the given figure, $O$ is a point inside $△PQR$ such that $\angle POR=90°$, $OP=6 \mathrm{cm}$ and $OR=8 \mathrm{cm}$. If $PQ=24 \mathrm{cm}$, $QR=26 \mathrm{cm}$. Prove that $△QPR$ is a right angled triangle.

In the given figure, $ABCD$ is a trapezium in which $AB\parallel DC$. The diagonals $AC$ and $DB$ intersect at $O$. Prove that $\frac{OA}{OC}=\frac{OB}{OD}$.

In figure, $\angle BAC=90°$, $AD\perp BC$. Prove that : $A{B}^2+C{D}^2=B{D}^2+A{C}^2$.

In $△ABC$, in the given figure, $PQ$ meets $AB$ in $P$ and $AC$ in $Q$. If $AP=1 \mathrm{cm}$, $PB=3 \mathrm{cm}$, $AQ=1.5 \mathrm{cm}$, $QC=4.5 \mathrm{cm}$, prove that area of $△APQ$ is one sixteenth of the area of $△ABC$.