NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)
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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:
i. and
ii.
iii. 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
i. In ,
- R is the midpoint of DC and S is the midpoint of DA.
- Thus by mid-point theorem, and
ii. In ,
- P is the midpoint of AB and Q is the midpoint of BC.
- Thus by mid-point theorem, and
- Also,
- Thus,
iii. - 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.
Thus, PQRS is a parallelogram.
Also,
Now, in PQRS,
- and from equation (1) and (2)
- Thus, PQRS is a rectangle.