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.

In the figure below, $\mathrm{PA}, \mathrm{QB}$ and $\mathrm{RC}$ are each perpendicular to $\mathrm{AC}$. If $x=8 \mathrm{cm}$ and $z=6 \mathrm{cm}$, then $y$ is equal to:

Sonali is standing on one side of a $7 \mathrm{m}$ wide road as shown below. She wants to estimate the distance $\left(D\right)$ between two light poles on the other side without crossing the road. 

Which of the following expressions represent $D$ in terms of$p$and $r$?

In the given figure, $AB\parallel CD$. Find $x$

Ravi is $1.82 \mathrm{m}$ tall. He wants to find the height of a tree in his backyard. From the tree's base he walked $12.20 \mathrm{m}$ along the tree's shadow to a position where the end of his shadow exactly overlaps the end of the tree’s shadow. He is now $6.10 \mathrm{m}$ from the end of the shadow. How tall is the tree? Express your answer in metre(m).