The lengths of the sides of a triangle are $13 \, cm$,$13 \, cm$,and $10 \, cm$. Find its area. Also,find the length of the altitude from the opposite vertex to the side whose length is $10 \, cm$.

In a parallelogram $ABCD$,the base $AB$ and the corresponding altitude $DM$ are $16 \, cm$ and $12 \, cm$,then the area of $ABCD =$ ................ $cm^{2}$.

The ratio of the measures of the sides of a triangle is $3: 4: 6$. If the semiperimeter of the triangle is $39 \, cm$,the measure of the longest side of the triangle is ............ $cm$.

State whether each of the following statements is true or false:
$(1)$ In a square,if the length of each side is $20 \, cm$,then the length of its diagonal is $20 \sqrt{2} \, cm$.
$(2)$ If each side of an equilateral triangle is doubled,the area of the triangle is twice the area of the original triangle.

Area of a square $= (........... )^{2}.$

There is a triangular party plot with side lengths of $200\,m$,$210\,m$,and $290\,m$. Calculate the cost of raising a lawn in that plot at the rate of Rs. $5/m^2$.