MATHEMATICAL REASONING

In the given parallelogram, find the length of the altitude from vertex $A$ on the side $DC$.

A rhombus shaped sheet with perimeter $40 \mathrm{cm}$ and one diagonal $12 \mathrm{cm},$ is painted on both sides at the rate of $₹5 \mathrm{per} {\mathrm{cm}}^2$. Find the cost of painting.

The area of a triangle, two sides of which are $8 \mathrm{cm}$ and $11 \mathrm{cm}$ and the perimeter is $32 \mathrm{cm}$is $k\sqrt{30 }{\mathrm{cm}}^2$. Find the value of k.

In the given figure, $∆\mathrm{ABC}$ has sides $\mathrm{AB}=7.5 \mathrm{cm},\mathrm{AC}=6.5 \mathrm{cm} \mathrm{and} \mathrm{BC}=7 \mathrm{cm}.$ On base $\mathrm{BC}$ a parallelogram $\mathrm{DBCE}$of same area as that of $∆\mathrm{ABC}$ is constructed. Find the height $\mathrm{DF}$ of the parallelogram.

The sides of a triangle are $11 \mathrm{cm},15 \mathrm{cm}$ and $16 \mathrm{cm}$ . The altitude to the largest side is ______.

A field is in the shape of a trapezium whose parallel sides are $25 \mathrm{m}$ and $10 \mathrm{m}$ . The non-parallel sides are $14 \mathrm{m}$ and $13 \mathrm{m}$. Find the area of the field.

Find the cost of laying grass in a triangular field of sides $50 \mathrm{m},65 \mathrm{m} \mathrm{and} 65 \mathrm{m}$ at the rate of ₹ $7 \mathrm{per} {\mathrm{m}}^2$.

The base of an isosceles triangle measures $24 \mathrm{cm}$ and its area is $192 {\mathrm{cm}}^2$. Find its perimeter.