Chapter Test

In the figure given below, $PQRS$ is a parallelogram formed by drawing lines parallel to the diagonals of a quadrilateral $\mathrm{ABCD}$ through its corners. Prove that area of the parallelogram $\mathrm{PQRS}=2\times$ area of the quadrilateral $\mathrm{ABCD}$.

In the parallelogram $ABCD, P$ is a point on the side $AB$ and $Q$ is a point on the side $BC$. Prove that the area of $∆\mathrm{CPD}=$ the area of $∆\mathrm{AQD}$.
 

In the parallelogram $ABCD, P$ is a point on the side $AB$ and $Q$ is a point on the side $BC$. Prove that the area of $∆\mathrm{ADQ}=$ the area of $∆\mathrm{APD}+$ area of $∆\mathrm{CPB}$.

In the given figure, $X$ and $Y$ are points on the side $LN$ of triangle $LMN$. Through $X$, a line is drawn parallel to $LM$ to meet $MN$ at $Z$. Prove that area of $∆\mathrm{LZY}=$ area of quadrilateral $MZYX$.

Perpendiculars are drawn from a point within an equilateral triangle to the three sides. Prove that the sum of the three perpendiculars is equal to the altitude of the triangle.

If each diagonal of a quadrilateral divides it into two triangle if equal areas, then prove that the quadrilateral is a parallelogram.

In the adjoining figure, $ABCD$ is a parallelogram in which $BC$ is produced to $E$ such that $CE=BC$. $AE$ intersects $CD \text{at} F$. If area of $∆DFB= 3 {\mathrm{cm}}^2$, find the area of parallelogram $ABCD$.

In the figure $X$ and $Y$ are the midpoints of $AC$ and $AB$ respectively, $QP\parallel BC$ and $CYQ$ and BXP is straight lines. Prove that $ar(∆ABP)=ar(∆ACQ)$.