If two arcs of a circle are congruent, then their corresponding chords are equal. Prove it.

Find the mean of the following numbers:

$$\begin{gathered} 16 14 12 8 23 26 18 20 \\ 9 11 22 27 18 20 23 \end{gathered}$$

State the value of the mean when each of these numbers is decreased by $6$.

The sides of right-angled triangle, containing the right angle are $5x \mathrm{cm}$ and $\left(3x-1\right) \mathrm{cm}$. Calculate the length of the hypotenuse of the triangle, if its area is $60 {\mathrm{cm}}^2$.

The following figure shows a right-angled triangle $ABC$ with $\angle B=90°, AB=15 \mathrm{cm}$ and $AC=25 \mathrm{cm}$. $D$ is a point in the side $BC$ and $CD=7 \mathrm{cm}$. If $DE\perp AC$, find the length of $DE$.

A small indoor green house(herbarium) is made entirely of glass panes including base, held together with tape. It is $30 \mathrm{cm}$ long, $25 \mathrm{cm}$ wide and $25 \mathrm{cm}$ height. How much tape is needed for all the $12$ edges?

In $△ABC, \angle C=90°, AB=20$ and $BC=12$. $D$ is a point in side $AC$ such that $CD=9$. Taking $\angle BDC=x$, find $\sin \angle ABC$.

In $△ABC, \angle C=90°, AB=20$ and $BC=12$. $D$ is a point in side $AC$ such that $CD=9$. Taking $\angle BDC=x$, find $\tan x-\cos x+3\sin x$.

In $△ABC, \angle B=90°, AB=40$ and $AC+BC=80$. Find $\sin A$.