**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**

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

**Materials Required**

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

**Procedure**

- Take any coloured glazed paper.
- Draw a quadrilateral of any dimensions on glazed paper and name it as ABCD.
- Cut that quadrilateral from the glazed paper.
- 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).

- Now, fold the figure along EF, GF, GH and EH. Press it and then unfold it as shown in fig. (ii).

- We will get creases along EF, GF, GH, HE.
- Make a replica (true copy) of EFGH (say PQRS) by using a tracing paper [fig.(iii)].

- Cut the quadrilateral PQRS along any diagonal (say RP) [fig.(iv)].

- We will get two triangles ∆PSR and ∆PQR.
- 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 ?

**Answer:**

A quadrilateral is a plane closed figure bounded by four line segments.

**Question 2.**

What are two main properties of a quadrilateral ?

**Answer:**

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

**Question 3.**

Is a parallelogram a quadrilateral ?

**Answer:**

Yes

**Question 4.**

Write two main properties of a parallelogram.

**Answer:**

- 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 ?

**Answer:**

Rectangle.

**Question 6.**

Do you know any difference between a parallelogram and a trapezium ?

**Answer:**

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 ?

**Answer:**

Base x corresponding altitude.

**Question 8.**

If base and altitude of a parallelogram are same, then what will be area of parallelogram ?

**Answer:**

Base x altitude.

**Question 9.**

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

**Answer:**

Square.

**Question 10.**

What do you mean by a parallelogram ?

**Answer:**

A parallelogram is a quadrilateral in which opposite sides are equal and parallel.

**Question 11.**

Write the name of different kinds of parallelograms.

**Answer:**

Rectangle, square and rhombus.

**Question 12.**

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

**Answer:**

Parallelogram.

**Question 13.**

Which theorem is used in this activity ?

**Answer:**

Mid-point theorem.

**Question 14.**

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

**Answer:**

Rhombus.

**Question 15.**

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

**Answer:**

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

**Answers**

- (ii)
- (i)
- (i)
- (ii)
- (iv)
- (iii)
- (i)
- (i)
- (i)
- (ii)

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