Prove that two triangles having the same base and equal areas lie between the same parallels.

The vertex $\mathrm{A}$ of a triangle $\mathrm{ABC}$ is joined to a point $\mathrm{D}$ on the side $\mathrm{BC}$. The midpoint of $\mathrm{AD}$ is $\mathrm{X}$ . Prove that the area of triangle $\mathrm{BXC}$ is half the area of triangle $\mathrm{ABC}$.

In the given figure, $\mathrm{E}$ is any point on median $\mathrm{A}\mathrm{D}$ of a $△\mathrm{A}\mathrm{B}\mathrm{C}$. Show that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{B}\mathrm{E}\right)=\mathrm{}\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{C}\mathrm{E}\right)$.

$\mathrm{D}$is the midpoint of side $\mathrm{BC}$ of $∆\mathrm{ABC}$ and $\mathrm{E}$ is the midpoint of $\mathrm{BD}$. If $\mathrm{O}$is the midpoint of $\mathrm{AE},$ prove that $\mathrm{ar}(∆\mathrm{BOE}) =\frac{1}{8} \mathrm{ar}(∆\mathrm{ABC}).$

In the given figure, the area of triangle $\mathrm{ABC}=7.2 {\mathrm{cm}}^2$, $\mathrm{CM}=\mathrm{MB}$ and $\mathrm{AL}=2\mathrm{LB}$. If the area of the triangle $\mathrm{ALM}$ is $k c{m}^2$ then find the value of $k$.

Prove that the parallelogram formed by joining the midpoints of the adjacent sides of a quadrilateral is half of the latter.

In the given figure, diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ of quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ intersect at $\mathrm{O}$ such that $\mathrm{O}\mathrm{B}=\mathrm{O}\mathrm{D}.$ If $\mathrm{A}\mathrm{B}=\mathrm{C}\mathrm{D},$ then show that 

$\mathrm{D}\mathrm{A}\parallel \mathrm{C}\mathrm{B}$ or $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram.

$\mathrm{D}$ and $\mathrm{E}$ are points on sides $\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C}$ respectively of $△\mathrm{A}\mathrm{B}\mathrm{C}$ such that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{D}\mathrm{B}\mathrm{C}\right)=\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{E}\mathrm{B}\mathrm{C}\right).$ Prove that $\mathrm{D}\mathrm{E}\parallel \mathrm{B}\mathrm{C}.$

An elastic belt is put around the pulley of a flour mill, of radius $3.5 \mathrm{cm}$, which again passes on the small pulley (of negligible radius). Find the area of the shaded region in the following figure. $\left[\mathrm{Use} \mathrm{\pi }=\frac{22}{7}\right]$  

If the area of the shaded region is $k {\mathrm{cm}}^2$, then find the value of $k$ (Correct up to two decimal places).

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

In the parallelogram $\mathrm{ABCD}$, the side $\mathrm{AB}$ is produced to$\mathrm{X}$, so that $\mathrm{BX}=\mathrm{AB}$. The line $\mathrm{DX}$cuts $\mathrm{BC}$ at $\mathrm{E}$. Prove that area $\mathrm{AED}=$twice the area $\mathrm{CEX}.$

Prove that a median divides a triangle into two triangles of equal area.

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is parallelogram and $\mathrm{B}\mathrm{C}$ is produced to a point $\mathrm{Q}$ such that $\mathrm{A}\mathrm{D}=\mathrm{C}\mathrm{Q}.$ If $\mathrm{A}\mathrm{Q}$ intersect $\mathrm{D}\mathrm{C}$ at $\mathrm{P},$ show that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{P}\mathrm{C}\right)=\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{D}\mathrm{P}\mathrm{Q}\right).$

The vertex $\mathrm{A}$of $∆ABC$ is joined to a point $\mathrm{D}$ on the side $\mathrm{BC}$. The midpoint of $\mathrm{AD}$ is $\mathrm{E}$.

Prove that $\mathrm{ar}(∆\mathrm{BEC})=\frac{1}{2}\mathrm{ar}(∆\mathrm{ABC}).$

In a triangle $\mathrm{A}\mathrm{B}\mathrm{C}, \mathrm{E}$ is the mid-point of median $\mathrm{A}\mathrm{D}.$ Show that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{E}\mathrm{D}\right)=\frac{1}{4}\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{B}\mathrm{C}\right)$

In figure, ABC is a triangle in which $D$ is the midpoint of  $BC$ and $E$ is the midpoint of $AD$. Prove that the area of$△BED=\frac{1}{4} area of △ABC$.