Exercise 6.2

(i) In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AB = 12\ \text{cm}$, $AC = 9\ \text{cm}$ and $AD = 4\ \text{cm}$, find $EC$.

(ii) In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AB = 8\ \text{cm}$, $BC = 6\ \text{cm}$ and $DE = 3\ \text{cm}$, find $AD\,.$

The diagonals of a quadrilateral $ABCD$ intersect each other at the point $O$ such that $\displaystyle \frac{AO}{CO}=\frac{BO}{DO}$. Show that $ABCD$ is a trapezium.

E and F are points on the sides PQ and PR respectively of a $\triangle$ PQR. For each of the following cases, state whether EF || QR :

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

In Fig. 6.18, if $LM \parallel CB$ and $LN \parallel CD$, prove that $\displaystyle \frac{AM}{AN} = \frac{AB}{AD} = \frac{LM}{LN}\,.$

In Fig. 6.19, $DE \parallel AC$ and $DF \parallel AE$. Prove that $BF \times BE = FE \times EC\,.$

In Fig. 6.20, $DE \parallel OQ$ and $DF \parallel OR$. Show that $EF \parallel QR\,.$

In Fig. 6.21, $A$, $B$ and $C$ are points on $OP$, $OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.

Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

$ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\dfrac{AO}{CO} = \dfrac{BO}{DO}$.