# The Triangle and Its Properties NCERT Solution for Class 7 Maths Chapter 6 Exercise 6.4

## The Triangle and Its Properties Exercise 6.4 NCERT Solution for Class 7 Maths Chapter 6

Exercise 6.4

Q.1:-  Is it possible to have a triangle with the following sides?

(i) 2cm, 3cm, 5cm
(ii) 3cm, 6cm, 7cm
(iii) 6cm, 3cm, 2cm

Solution 1:-

Concept:- A triangle is possible if sum of the lengths of any two sides would be greater
than the length of third side.

Solution 1(i):- Let us check this

2 + 3 > 5      No
2 + 5 > 7     Yes
5 + 7 > 2      Yes

This Triangle is not Possible.

Solution 1(ii):- Let us check this

3 + 6 > 7      Yes
3 + 7 > 6     Yes
5 + 7 > 3      Yes

This Triangle is Possible.

Solution 1(iii):- Let us check this

2 + 3 > 6      No
2 + 6 > 3     Yes
3 + 6 > 2      Yes

This Triangle is not Possible.

Q.2:- Take any point O in the interior of a triangle PQR. Is

(i) OP + OQ > PQ
(ii) OQ + OR > QR
(iii) OR + OP > RP

Solution 2(i):- Join OR,OP and OQ

Is OP+OQ > PQ ?
Yes, because POQ form a triangle.

Solution 2(ii):- Is OQ+OR > QR ?
Yes, because QOR form a triangle.

Solution 2(iii):- Is OR+OP > RP ?
Yes, because POR form a triangle.

Q.3:- AM is a median of a triangle ABC.
Is AB + BC + CA > 2 AM?
(Consider the sides of triangles
ABM and AMC.)

Solution 3:- Since, the sum of lengths of any two sides in a triangle should be greater than the length
of third side.

In Triangle ABM,     AB + BM > AM                 ……… equation (i)
In Triangle AMC,    AC + MC > AM                ………. equation (ii)

By adding equation (i) + (ii)
AB + BM + AC + MC > AM + AM
AB + AC + ( BM + MC ) > 2AM
AB + AC + BC > 2 AM

Hence, Yes it is possible.

Is AB + BC + CD + DA > AC + BD?

Solution 4:- Since, the sum of lengths of any two sides in a triangle should be greater than the length
of third side.

In Triangle ABC,       AB + BC > AC           ……. equation (i)
In Triangle DCB,       DC + CB > BD          …….equation (iii)
In Triangle DBA,       AB + AD > BD           …..equation (iv)

Adding equations (i), (ii), (iii) and (iv)

AB + BC + AD + DC + DC + CB + AB + AD > AC + AC + BD + BD
(AB + AB) + (BC + CB) + (AD + AD) + (DC + DC) > 2AC + 2BD
2AB + 2BC + 2AD + 2DC > 2(AC + BD)
2 (AB+BC+AD+DC) > 2(AC + BD)
AB + BC + CD + DA > AC + BD

Yes, it is True/Possible.

AB + BC + CD + DA < 2 (AC + BD)?

Solution 5:-

Since, the sum of lengths of any two sides in a triangle should be greater than the length
of third side.

In Triangle AOB,       AB < OA + OB           ……. equation (i)
In Triangle BOC,       BC < OB + OC          ……..equation (ii)
In Triangle COD,       CD < OC + OD          …….equation (iii)
In Triangle DOA,       DA < OA + OD           …..equation (iv)

Adding equations (i), (ii), (iii) and (iv)

AB + BC + CD + DA < OA + OB + OB + OC + OC + OD + OA + OD,
AB + BC + CD + DA < (OA + OA) + (OB + OB) + (OC + OC) + (OD + OD),
AB + BC + CD + DA < 2OA + 2OB + 2OC + 2OD,
AB + BC + CD + DA < 2(OA + OB + OC + OD),
AB + BC + CD + DA < 2[(OA + OC) + (OB + OD)]

AB + BC + CD + DA < 2(AC + BD)

Yes, it is True/Possible.

Q.6:- The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall ?

Solution 6:-

We know that the sum of the two sides of a triangle is always greater than the third side.

Therefore, third side has to be less than the sum of two side,

Thus the third side should be less than (12cm + 15cm) = 27cm

The side cannot be less than the difference of the two sides.

Thus the third side should be more than (15cm – 12cm) = 3cm

The length of the third side could be any length greater than 3 and less than 27cm.

Answer :- Between 3cm and 27cm.

NCERT Solutions for Class 7 Math, Chapter 6 – The Triangle and Its Properties, Exercise-6.4