NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4

Class 7 Maths NCERT Solutions Chapter 6 The Triangle and its Properties Ex 6.4

Ex 6.4 Class 7 Maths Question 1.
Is it possible to have a triangle with the following sides?
(i) 2 cm, 3 cm, 5 cm
(ii) 3 cm, 6 cm, 7 cm
(iii) 6 cm, 3 cm, 2 cm
Solution:
(i) 2 cm, 3 cm, S cm We have 2 + 3 = 5
⇒ Sum of the lengths of two sides = Length of the third side
This is impossible since the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

(ii) 3 cm, 6 cm, 7 cm
We see that 3 + 6 > 7
6 + 7 > 3
7 + 3 > 6
Therefore, it is possible to have a triangle with side lengths 3 cm, 6 cm, 7 cm.

(iii) 6 cm, 3 cm, 2 cm
We see that 6 + 3 = 9 > 2
3 + 2 = 5 \(\ngtr \) 6
2 + 6 = 8 > 3
Therefore, it is not possible to have a triangle with side lengths 6 cm, 3 cm, 2 cm.

Ex 6.4 Class 7 Maths Question 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:
(i) Yes, OP + OQ > PQ because on joining OP and OQ, we get an ∆OPQ and in a triangle, the sum of the lengths of any two sides is always greater than the third side.

(ii) Yes, OQ + OR > QR, because on joining OQ and
OR, we get an ∆OQR and in a triangle, the sum of the length of any two sides is always greater than the third side.

(iii) Yes, OR + OP > RP, because on joining OR and OP, we get an ∆OPR and in a triangle, the sum of the lengths of any two sides is always greater than the third side.

Ex 6.4 Class 7 Maths Question 3.
AM is median of a triangle ABC. Is AB + BC + CA > 2AM?
(Consider the sides of triangles ∆ABM and ∆AMC.)

Solution:
In ∆ ABM, AB + BM > AM …(1)
The Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ ACM,
CA + CM > AM …(2)
The Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Sum (1) and (2),
(AB + BM) + (CA + CM) > AM + AM
⇒ AB + (BM + CM) + CA > 2 AM
⇒ AB + BC + CA > 2 AM.

Ex 6.4 Class 7 Maths Question 4.
ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

Solution:

Ex 6.4 Class 7 Maths Question 5.
ABCD is a quadrilateral. Is AB + BC + CD + DA < 2(AC + BD)?
Solution:
Let ABCD be a quadrilateral and its diagonals AC and BD intersect at O. Using triangle inequality property, we have
In ∆OAB
OA + OB > AB ……(1)

Ex 6.4 Class 7 Maths Question 6.
The lengths of the two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?
Solution:
Let x cm be the length of the third side.
Thus, 12 + 15 > x, x + 12 > 15 and x + 15 > 12
⇒ 27 > x, x > 3 and x > -3
The numbers between 3 and 27 satisfy these.
∴ The length of the third side could be any length between 3 cm and 27 cm.