R S Aggarwal and Veena Aggarwal Solutions for Exercise 3: EXERCISE 10C

In a parallelogram $PQRS,$ $PQ=12 \mathrm{cm}$ and $PS=9 \mathrm{cm}$. The bisector of $\angle P$ meets $SR$ in $M. PM$ and $QR$ both when produced meet at $T$. If the length of $RT=k \mathrm{cm}$, then find the value of $k$.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rectangle is a rhombus.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a square is a square.

Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

The diagonals of a quadrilateral $ABCD$ are equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus.

The diagonals of a quadrilateral $ABCD$ are perpendicular to each other. Prove that the quadrilateral formed by joining the midpoints of its sides is a rectangle.

The midpoints of the sides $AB, BC, CD$ and $DA$ of a quadrilateral $ABCD$ are joined to form a quadrilateral. If $AC=BD$ and $AC\perp BD$, then prove that the quadrilateral formed is a square.