In the figure,$DE \parallel OQ$ and $DF \parallel OR$. Show that $EF \parallel QR$.

Nazima is fly fishing in a stream. The tip of her fishing rod is $1.8 \, m$ above the surface of the water and the fly at the end of the string rests on the water $2.4 \, m$ away from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut,how much string does she have out? If she pulls in the string at the rate of $5 \, cm$ per second,what will be the horizontal distance of the fly from her after $12 \, seconds$?

$ABC$ is an isosceles triangle right-angled at $C$. Prove that $AB^{2} = 2AC^{2}$.

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.

$D$ is a point on the side $BC$ of a triangle $ABC$ such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB \cdot CD$.

$BL$ and $CM$ are medians of a triangle $ABC$ right-angled at $A$. Prove that $4(BL^2 + CM^2) = 5BC^2$.