In the figure below,$OPQ$  is a triangle with $OP=OQ$. $RS$ is an arc of a circle with centre $O$.

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Triangle $OSR$ is similar to triangle $OPQ$.

Is the above statement true or false? Justify your reason 

Are the two quadrilaterals shown below similar? Give a reason for your answer.

Looking at the above figure, Hari said that the shortest distance between Town $E$ and Town $F$ is $15$ km.

Is the statement True or False? Justify your answer.

Shown below is a rectangle $PQRS$. $\angle UTQ$ is a right angle.

Which of these is TRUE about $∆UST$ and $∆TRQ$?

Leela has a triangular cabinet that fits under his staircase. There are four parallel shelves as shown below.

The total height of the cabinet is $144 \mathrm{cm}$. What is the maximum height of a book that can stand upright on the bottom-most shelf?

In the figure below, $ST \left|\right| PQ$. All measurements are in units.

Prove that the area of the trapezium$\text{ PQTS is }\left[\frac{1}{2}(a+b)h\right]\text{ sq units}$.

Just as in mathematics, principles in physics often lead to other useful results. Fermat's principle states that light travels in a path that requires the least time. How light behaves when reflected off a surface (called the law of reflection) can be derived from this principle. The law of reflection states that when light hits a reflective surface at an angle, it will reflect off that surface at the same angle.

Describe how this information could be used to find the height of the lamppost.