R Chandra Solutions for Chapter: Areas of Parallelograms and Triangles, Exercise 1: TEST YOUR SKILLS

In the figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram and $\mathrm{E}\mathrm{F}\mathrm{C}\mathrm{B}$ is a rectangle. Also $\mathrm{A}\mathrm{L}\perp \mathrm{B}\mathrm{C}$.

Prove that: $\mathrm{}\mathrm{a}\mathrm{r}\left(\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\right)=\mathrm{a}\mathrm{r}\left(\mathrm{E}\mathrm{F}\mathrm{C}\mathrm{B}\right)$

In the figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram and $\mathrm{E}\mathrm{F}\mathrm{C}\mathrm{B}$ is a rectangle. Also $\mathrm{A}\mathrm{L}\perp \mathrm{B}\mathrm{C}$. Prove that:
 
$\mathrm{a}\mathrm{r}\left(\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}=\mathrm{a}\mathrm{r}\left(\mathrm{BC}\times \mathrm{A}\mathrm{L}\right)\right)$

In the adjoining figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram such that $\mathrm{A}\mathrm{E}\perp \mathrm{CD}$ and $\mathrm{}\mathrm{C}\mathrm{F}\perp \mathrm{A}\mathrm{D}.$
If $\mathrm{A}\mathrm{B}=10\mathrm{}\mathrm{c}\mathrm{m},\mathrm{}\mathrm{A}\mathrm{E}=6\mathrm{}\mathrm{c}\mathrm{m}$ and $\mathrm{C}\mathrm{F}=8\mathrm{}\mathrm{c}\mathrm{m},$ then find $\mathrm{A}\mathrm{D}$.

In the figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a quadrilateral and $\mathrm{B}\mathrm{E}\parallel \mathrm{A}\mathrm{C}$ and also $\mathrm{B}\mathrm{E}$ meets $\mathrm{D}\mathrm{C}$ produced at $\mathrm{E}$. Show that area of $∆\mathrm{A}\mathrm{D}\mathrm{E}$ is equal to the area of the quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$.

$\mathrm{A}\mathrm{B}$ and $\mathrm{C}\mathrm{D}$ are parallel sides of a trapezium $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$. Diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ intersect at $\mathrm{O}$. Prove that:
$\mathrm{a}\mathrm{r}\left(∆\mathrm{A}\mathrm{O}\mathrm{D}\right)=\mathrm{a}\mathrm{r}(∆\mathrm{B}\mathrm{O}\mathrm{C})$.

If $D$ is midpoint of $AB$ of $∆ABC$ and $P,Q$ are the points on $BC$and $AB$ respectively. Such that $PD\parallel CQ$. Then show that $ar\left(∆BPQ\right)=\frac{1}{2}ar(∆ABC)$?

A rectangle is formed by joining the midpoints of the sides of a rhombus. Show that the area of rectangle is half the area of the rhombus.

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram. Through $\mathrm{A}\mathrm{},\mathrm{}$ line $\mathrm{A}\mathrm{E}\mathrm{F}$ is drawn to meet $\mathrm{B}\mathrm{C}$ at $\mathrm{E}$ and $\mathrm{D}\mathrm{C}$ produced at $\mathrm{F}$. Show that $\mathrm{a}\mathrm{r}(∆\mathrm{B}\mathrm{E}\mathrm{F})=\mathrm{a}\mathrm{r}(∆\mathrm{D}\mathrm{C}\mathrm{E})$.