Shown below is a map of Giri's neighbourhood.

Giri did a survey of his neighbourhood and collected the following information.
* The hotel, mall and the main gate of the garden lie in a straight line.
* The distance between the hotel and the mall is half the distance between the mall and the main gate of the garden.
* The bus stand is exactly midway between the main gate of the garden and the fire station.
* The mall, bus stand and the water tank lie in a straight line.

What is the $x$-coordinate of the mall's location?

Find the distance between the following pair of points.

$\left(-5,7\right), \left(-1,3\right)$

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

$\left(4,\mathrm{}5\right),\mathrm{}\left(7,\mathrm{}6\right),\mathrm{}\left(4,\mathrm{}3\right),\mathrm{}\left(1,\mathrm{}2\right)$

$\Delta ABC$ with vertices $A(-2,0), B(2,0)$ and $C(0,2)$ is similar to $\Delta DEF$ with vertices $D(-4,0),E(4,0)$ and $F(0,4)$.

In sports day activities of Meerut Public School, the lines were drawn with a chalk powder in a rectangular shape $OBCD$. Each line is at $\frac{1}{2 } \mathrm{m}$ distance from each other. $60$ flower pots have been placed at a distance of $\frac{1}{2 } \mathrm{m}$ from each other along $OD$. Yamini runs ${\frac{1}{4}}^{\mathrm{th}}$ of the distance $OD$ on the $3^{\mathrm{rd}}$ line makes a red flower. Kamla runs ${\frac{1}{5}}^{\mathrm{th}}$ of the distance $OD$ on the $7^{\mathrm{th}}$ line and makes a yellow flower.

If the distance between these flowers is $k \mathrm{m}$, then find the value of $k$ in decimal form.

Show that the points $A(5, 6), B(1, 5), C(2, 1)$ and $D(6,2)$ are the vertices of a square.

Find the value of $\mathrm{a}$, if the distance between the points $\mathrm{A}\left(-3,-14\right)$ and $\mathrm{B}\left(\mathrm{a},-5\right)$ is $9$ units.