$ABCD$ is a trapezium in which $AB \| DC$, $BD$ is a diagonal and $E$ is the mid-point of $AD$. A line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$ (see Fig. 8.21). Show that $F$ is the mid-point of $BC$.

$ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that (iii) $CM = MA = \frac{1}{2}AB$

$ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that (ii) $MD \perp AC$

$ABCD$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ (see Fig 8.20). $AC$ is a diagonal. Show that : (iii) $PQRS$ is a parallelogram.

$ABCD$ is a rhombus and $P, Q, R$ and $S$ are the mid-points of the sides $AB, BC, CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rectangle.

$ABCD$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ (see Fig 8.20). $AC$ is a diagonal. Show that : (ii) $PQ = SR$

$ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that (i) $D$ is the mid-point of $AC$

$ABCD$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ (see Fig 8.20). $AC$ is a diagonal. Show that : (i) $SR \| AC$ and $SR = \frac{1}{2} AC$

In a parallelogram $ABCD$, $E$ and $F$ are the mid-points of sides $AB$ and $CD$ respectively (see Fig. 8.22). Show that the line segments $AF$ and $EC$ trisect the diagonal $BD$.

$ABCD$ is a rectangle and $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rhombus.