Q. $P$ and $Q$ are any two points lying on the sides $DC$ and $AD$ respectively of a parallelogram $ABCD$. Show that $area\left(\Delta APB\right)=area\left(\Delta BQC\right)$

Q. In the given fig, $ar\left(DRC\right)=ar\left(DPC\right)$ and $ar\left(BDP\right)=ar\left(ARC\right)$. Show that both the quadrilaterals $ABCD$ and $DCPR$ are trapeziums.

Q. In fig, $ABCDE$ is a pentagon. A line through $B$ parallel to $AC$ meets $DC$ produced at $F$. Show that
(i) $ar\left(ACB\right)=ar\left(ACF\right)$
(ii) $ar\left(AEDF\right)=ar\left(ABCDE\right)$

Q. $ABCD$ is a trapezium with $AB\parallel DC$. A line parallel to $AC$ intersects $AB$ at $X$ and $BC$ at $Y$. Prove that $ar\left(△ADX\right)=ar\left(△ACY\right)$

Q. In given figure, $E$ is any point on median $AD$ of a $△ABC$. Show that $area\left(ABE\right)=area\left(ACE\right)$.

Q. A farmer was having a field in the form of a parallelogram $PQRS$. She took any point A on $RS$ and joined it to points $P$ and $Q$. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

Q. Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB\parallel DC$ intersect each other at $O$. Prove that $ar\left(AOD\right)=ar\left(BOC\right)$

Q. $D,E$ and $F$ are respectively the mid-points of the sides $BC,CA$ and $AB$ of a $△ABC$. Show that
(i) $BDEF$ is a parallelogram
(ii) $ar\left(DEF\right)=\frac{1}{4}ar\left(ABC\right)$
(iii) $ar\left(BDEF\right)=\frac{1}{2}ar\left(ABC\right)$

Q. A villager Itwarri has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwarri agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented

Q. In a triangle $ABC,E$ is the mid-point of median $AD$. Show that $area\left(BED\right)=\frac{1}{4}area\left(ABC\right)$.

Q. Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels

Q. If $E,F,G$ and $H$ are respectively the mid-points of the sides of a parallelogram $ABCD$, show that $ar\left(EFGH\right)=\frac{1}{2}ar\left(ABCD\right)$

Q. Diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$ in such a way that $ar\left(△AOD\right)=ar\left(△BOC\right)$. Prove that $ABCD$ is a trapezium

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

Q. In the given Figure, $ABCD$ is a parallelogram, $AE\perp DC$ and $CF\perp AD$. If $AB=16cm,AE=8cm$ and $CF=10cm$, find $AD$.

Q. In Fig, $AP\parallel BQ\parallel CR$. Prove that $ar\left(AQC\right)=ar\left(PBR\right)$

Q. In the given Figure, diagonals $AC$ and $BD$ of quadrilateral $ABCD$ intersect at $O$ such that $OB=OD$. If $AB=CD$, then show that :
(i) $ar\left(DOC\right)=ar\left(AOB\right)$
(ii) $ar\left(DCB\right)=ar\left(ACB\right)$
(iii) $DA\parallel CB$ or $ABCD$ is a parallelogram.

Q. In fig, $PQRS$ and $ABRS$ are parallelogram and $X$ is any point on side $BR$. Show that

Q. In fig, $ABC$ and $ABD$ are two triangles on the same base $AB$. If line-segment $CD$ is bisected by $AB$ at $O$, show that $ar\left(ABC\right)=ar\left(ABD\right)$.

Q. $D$ and $E$ are points on sides $AB$ and $AC$ respectively of $△ABC$ such that $area\left(DBC\right)=area\left(EBC\right)$. Prove that $DE\parallel BC$