Exercise 6.2

In Fig. $\mathrm{D}\mathrm{E}\parallel \mathrm{B}\mathrm{C}$. Find  $\mathrm{AD}$ 

$\mathrm{E}\mathrm{}$ and $\mathrm{F}$ are points on the sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{R}$ respectively of a $△\mathrm{P}\mathrm{Q}\mathrm{R}$. State whether $\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$ where $\mathrm{PE}=3.9 \mathrm{cm}, \mathrm{EQ}=3 \mathrm{cm}, \mathrm{PF}=3.6 \mathrm{cm}$ and $\mathrm{FR}=2.4 \mathrm{cm}$.

$\mathrm{E}\mathrm{}$ and $\mathrm{F}$ are points on the sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{R}$ respectively of a $∆\mathrm{P}\mathrm{Q}\mathrm{R}$. For each of the following cases, state whether $\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$:

$PQ=1.28 \mathrm{cm}, PR=2.56 \mathrm{cm}, PE=0.18 \mathrm{cm}$ and $PF=0.36 \mathrm{cm}$

In the figure, $\mathrm{A},\mathrm{}\mathrm{B}$ and $\mathrm{}\mathrm{C}$ are points on $\mathrm{}\mathrm{O}\mathrm{P},\mathrm{}\mathrm{O}\mathrm{Q}$ and $\mathrm{}\mathrm{O}\mathrm{R}$ respectively such that $\mathrm{}\mathrm{A}\mathrm{B}\parallel \mathrm{P}\mathrm{Q}$ and $\mathrm{}\mathrm{A}\mathrm{C}\parallel \mathrm{P}\mathrm{R}$. Show that $\mathrm{B}\mathrm{C}\parallel \mathrm{Q}\mathrm{R}$.

Using Basic Proportionality Theorem (BPT) , prove that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class $\mathrm{I}\mathrm{X}$).

Using the converse of Basic Proportionality Theorem (BPT), prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a trapezium in which $\mathrm{}\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C}$ and its diagonals intersect each other at the point $\mathrm{O}$. Show that
$\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}$

The diagonals of a quadrilateral $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ intersect each other at the point $\mathrm{}\mathrm{O}$ such that $\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}$. Show that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a trapezium.