NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 1
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Q1 ABCD is a quadrilateral in which P, Q, R and S are midpoints 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 midpoints of the sides AB, BC, CD and DA

In ,

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

Thus by midpoint theorem, and


In,

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

Thus by midpoint theorem, and

Also,

Thus,


 From (i)

And,  from (ii)

 From (i) and (ii)

Also,

Thus, PQRS is a parallelogram.
Q2 ABCD is a rhombus and P, Q, R and S are the midpoints 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 midpoints 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 SideAngleSide 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 SideAngleSide 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.


Thus, PQRS is a parallelogram.
Also,
Now, in PQRS,

and from equation (1) and (2)

Thus, PQRS is a rectangle.