Manipur Board Solutions for Chapter: Area, Exercise 2: EXERCISE 9.2

If the diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ are perpendicular to one another, prove that area of the quadrilateral $=\frac{1}{2}\times AC\times BD$.

If the medians $AD$ and $BE$ of a triangle $ABC$ intersect at $O$, prove that area of $∆AOB=$ area of quadrilateral $CDOE$.

$ABCD$ is a parallelogram; $E$ and $F$ are the midpoints of the sides $BC$ and $CD$. Prove that area of $∆AEF=\frac{3}{8}$ (area of parallelogram $ABCD$).

If each diagonal of a quadrilateral divides it into two triangles of equal area, show that the quadrilateral is a parallelogram.

Triangles $ABC$ and $DBC$ are on the same base $BC$ and on the opposite sides of $BC$ such that area of $∆ABC=$ area of $∆DBC$. Show that $BC$ bisects $AD$.

Any point $D$ is taken in the base $BC$ of $∆ABC$ and $AD$ is produced to $E$ such that $AD=DE$. Show that $\mathrm{area} \mathrm{of}∆BCE=\mathrm{area} \mathrm{of} ∆ABC$.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$ and divide the quadrilateral $ABCD$ into four triangles of equal area. Show that $ABCD$ is a parallelogram.

If the medians of a $∆ABC$ intersect at $O$, prove that $\mathrm{ar}\left(∆AOB\right)=\mathrm{ar}(∆BOC)=\mathrm{ar}(∆COA)=\frac{1}{3}\times \mathrm{ar}\left(ABC\right)$