## CBSE Class 9 Maths Lab Manual – Quadrilateral Formed by Joining Mid-points of Sides of a Quadrilateral

Objective
To show that the quadrilateral formed by joining the mid-points of the adjacent sides of a quadrilateral is a parallelogram by paper folding.

Prerequisite Knowledge

1. Concept of finding mid-point of a line segment by performing paper folding activity.
2. Properties of a parallelogram.

Materials Required
Glazed papers, pencil, a pair of scissors, gluestick and tracing paper.

Procedure

1. Take any coloured glazed paper.
2. Draw a quadrilateral of any dimensions on glazed paper and name it as ABCD.
3. Cut that quadrilateral from the glazed paper.
4. Now, find the mid-point of each side AB, BC, CD, DA by paper folding and name them E, F, G, H respectively as shown infig. (i). 5. Now, fold the figure along EF, GF, GH and EH. Press it and then unfold it as shown in fig. (ii). 6. We will get creases along EF, GF, GH, HE.
7. Make a replica (true copy) of EFGH (say PQRS) by using a tracing paper [fig.(iii)]. 8. Cut the quadrilateral PQRS along any diagonal (say RP) [fig.(iv)]. 9. We will get two triangles ∆PSR and ∆PQR.
10. Now, overlap these two triangles. Two triangles coincide with each other [fig.(v)] such that side PS overlaps with QR and PQ with SR. Observation
We observe that two triangles coincide with each other which means two triangles are congruent to each other. In a quadrilateral, two triangles cover each other completelv along any diagonal, then the quadrilateral will be a parallelogram.
∴ ∆PQR = ∆PSR
i.e., ar (∆PQR) = ar(∆PSR)
∴ PQRS is a parallelogram.

Result
As the replica of ∆PQR exactly covers the replica of ∆PSR
∴ PQ = RS, QR=SP
∴ PQRS is a parallelogram.

Learning Outcome
We have verified by paper folding that the quadrilateral formed by joining the mid-points of adjacent sides of a quadrilateral will be a parallelogram. We also learnt that a diagonal always divides the parallelogram into two triangles of equal areas.

Activity Time
What type of figures do you obtain?

• If you join mid-points of the sides of a rectangle (Do it by paper folding).
• If you join the mid-points of thesidesof a square (Do it by paper folding).

Viva Voce

Question 1.
What do you mean by a quadrilateral ?
A quadrilateral is a plane closed figure bounded by four line segments.

Question 2.
What are two main properties of a quadrilateral ?

• Sum of four angles is 360°.
• It has 4 sides.

Question 3.
Is a parallelogram a quadrilateral ?
Yes

Question 4.
Write two main properties of a parallelogram.

• Diagonals bisect each other.
• Opposite sides are equal.

Question 5.
In a parallelogram, if one angle is 90°, then what type of parallelogram you will get ?
Rectangle.

Question 6.
Do you know any difference between a parallelogram and a trapezium ?
In a parallelogram, two pairs of opposite sides are parallel. In a trapezium, one pair of opposite sides is parallel.

Question 7.
What is the area of a parallelogram ?
Base x corresponding altitude.

Question 8.
If base and altitude of a parallelogram are same, then what will be area of parallelogram ?
Base x altitude.

Question 9.
If base and altitude of a parallelogram are same, then what type of parallelogram will be obtained ?
Square.

Question 10.
What do you mean by a parallelogram ?
A parallelogram is a quadrilateral in which opposite sides are equal and parallel.

Question 11.
Write the name of different kinds of parallelograms.
Rectangle, square and rhombus.

Question 12.
If you join the mid-points of consecutive sides of a quadrilateral, what shape will you obtain ?
Parallelogram.

Question 13.
Which theorem is used in this activity ?
Mid-point theorem.

Question 14.
If you join the mid-points of consecutive sides of a rectangle, what figure will you obtain ?
Rhombus.

Question 15.
If you join the mid-points of consecutive sides of a rhombus, what figure will you obtain ?
Rectangle.

Multiple Choice Questions

Question 1.
Name the quadrilateral formed by joining the mid¬points of the consecutive sides of a square:
(i) rectangle
(ii) square
(iii) rhombus
(iv) none of these

Question 2.
The four triangles formed by joining the mid-points of three sides of a triangle are:
(i) congruent
(ii) non-congruent
(iii) similar
(iv) none of these

Question 3.
In the given figure ABCD, if P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively, then: (i) SR = $$\frac { 1 }{ 2 }$$ AC
(ii) SR = AC
(iii) SR = $$\frac { 1 }{ 3 }$$ AC
(iv) none of these

Question 4.
In ∆ABC, if E is the mid-point of AC, F lies on BC and EF // AB then:
(i) EF = $$\frac { 1 }{ 3 }$$ AB
(ii) EF = $$\frac { 1 }{ 2 }$$ AB
(iii) EF = AB
(iv) none of these

Question 5.
In a parallelogram, the figure formed by joining the mid-points of consecutive sides is :
(i) a rectangle
(ii) a rhombus
(iii) a square
(iv) none of these.

Question 6.
In a rhombus, diagonals bisect each other at an angle of:
(i) 45° and 135°
(ii) 60° and 120°
(iii) 90°
(iv) none of these

Question 7.
In a rectangle, diagonals are:
(i) equal
(ii) not equal
(iii) half of each other
(iv) none of these

Question 8.
The straight line joining the mid-points of the non¬parallel sides of a trapezium is parallel to:
(i) parallel sides
(ii) non-parallel sides
(iii) one non-parallel side
(iv) none of these

Question 9.
The triangle formed by joining the mid-points of the sides of a right triangle is :
(i) a right triangle
(ii) an obtuse-angled triangle
(iii) an isosceles triangle
(iv) none of these

Question 10.
The triangle formed by joing the mid-points of the sides of an isosceles triangle is:
(i) an equilateral triangle
(ii) an isosceles triangle
(iii) a right-angled triangle
(iv) none of these