NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 1

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Quadiletrals and the properties of their diagonals

Diagonals of a Quadiletrals

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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.

ABCD is a Quadrilateral

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:

ABCD is Rhombus and PQRS is Rectangle

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.