Prove that the rectangle circumscribing a circle is a square.

In adjoining figure, $\mathrm{DEFG}$ is a square and $\angle \mathrm{BAC}=90°$, show that ${\mathrm{FG}}^2=\mathrm{BG}\times \mathrm{FC}$.

Isha is $10 years$ old girl. On the result day, Isha and her father Suresh were very happy as she got first position in the class. While coming back to their home, Isha asked for a treat from her father a a reward for her success. They went to a juice shop and asked for two glasses of juice. Aisha, a juice seller was serving juices to her customers in two type of glasses. Both the glasses has inner radius $3 cm$. The height of both the glasses was $10 cm$. 

First type: A glass with hemispherical raised bottom.

Second type: A glass with conical raised bottom of height $1.5 cm$. Isha insisted to have the juice in first type of glass and her father decided to have the juice in second type of glass. Out of the two, Isha or her father Suresh, who got more quantity of juice to drink and by how much?

If the sum of the first $m$ terms of an $AP$ is the same as the sum of its first $n$ terms, show that the sum of its first $\left(m+n\right)$ terms is zero.

In the below given figure, $ABCDE$ is a pentagon with $BE\parallel CD$ and $BC\parallel DE$. $BC$ is perpendicular to $CD$. $AB=5 cm, AE=5 cm, BE=7 cm, BC=x-y$ and $CD=x+y$. If the perimeter of $ABCDE$ is $27 cm$.Find the value of $x$ and $y$, given $x, y\ne 0$.

Solve the following system of equations;

$\frac{21}{x}+\frac{47}{y}=110$

$\frac{47}{x}+\frac{21}{y}=162, x, y\ne 0$