Q-1 ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig). AC is a diagonal. Show that:

and

PQRS is a parallelogram.

Solution:

Give, ABCD is a quadrilateral

P, Q, R and S are mid-points of the sides AB, BC, CD and DA

In ,

R is the midpoint of DC and S is the midpoint of DA.

Thus by mid-point theorem, and

In,

P is the midpoint of AB and Q is the midpoint of BC.

Thus by mid-point theorem, and

Also,

Thus,

- From (i)

And, - from (ii)

- From (i) and (ii)

Also,

Thus, PQRS is a parallelogram.

Q-2 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.

Solution:

Given,

ABCD is a rhombus

P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

Construction, AC and BD are joined.

Proof,

In and ,

(Halves of the opposite sides of the rhombus)

(Opposite angles of the rhombus)

(Halves of the opposite sides of the rhombus)

Thus, by Side-Angle-Side congruence condition.

By Corresponding Parts of Congruent Triangles ……equation (1)

In ΔQCR and ΔSAP,

(Halves of the opposite sides of the rhombus)

(Opposite angles of the rhombus)

(Halves of the opposite sides of the rhombus)

Thus, by Side-Angle-Side congruence condition.

By Corresponding Parts of Congruent Triangles ……equation (2)

Now, on

R and Q are the mid points of CD and BC respectively

Also,

P and S are the mid points of AD and AB respectively.