State Pythagoras theorem and its converse.

The lengths of the diagonals of a rhombus are $30 \mathrm{cm}$ and $40 \mathrm{cm}.$ Find the side of the rhombus in $\mathrm{cm}$. (Write the numeral value as final answer i.e without unit.)

In Fig., $PQ\Vert BC$ and $AP:PB=1:2$. If $\frac{\mathrm{area}\left(△APQ\right)}{area\left(△ABC\right)}=\frac{1}{k}$, find $k$.

In figure, $\mathrm{S}$ and $\mathrm{T}$ are points on the sides $\mathrm{PQ}$ and $\mathrm{PR}$, respectively of $△\mathrm{PQR}$ such that  $\mathrm{PT}=2 \mathrm{cm}, \mathrm{TR}=4 \mathrm{cm}$ and $\mathrm{ST}$ is parallel to $\mathrm{QR}$. Determine the ratio of the areas of $△\mathrm{PST}$ and $△\mathrm{PQR}$ and write the sum of their ratios as final solution.

In the figure, $△AHK$ is similar to $△ABC$, where $AK=10 \mathrm{cm}, BC=3.5 \mathrm{cm}$ and $HK=7 \mathrm{cm}$. If $AC$ is $k \mathrm{cm}$, then find the value of $k$. 

Answer the following question either in one word or one sentence or as per the requirement of the question:

In the figure, $\mathrm{DE}\parallel \mathrm{BC}$ in $△\mathrm{ABC}$ such that $\mathrm{BC}=8 \mathrm{cm}, \mathrm{AB}=6 \mathrm{cm}$ and $\mathrm{DA}=1.5 \mathrm{cm}$. If $\mathrm{DE}=k \mathrm{cm}$, then write the value of $k$ as final answer.

Answer the following question either in one word or one sentence or as per the requirement of the question:

In the given figure, $DE\parallel BC$, $AD$ $=\frac{1}{2}\mathrm{BD}$ and $BC=4.5 \mathrm{cm}$. If $DE=k \mathrm{cm}$, then write the value of $k$ as final answer.

In Fig., $\mathrm{LMK}=\mathrm{PNK}=46°$. Express $x$ in terms of $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are lengths of $\mathrm{LM},$ $\mathrm{MN}$ and $\mathrm{NK}$, respectively and write its numerator.