NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4.

Board CBSE
Textbook NCERT
Class Class 7
Subject Maths
Chapter Chapter 6
Chapter Name The Triangle and its Properties
Exercise Ex 6.4
Number of Questions Solved 6
Category NCERT Solutions

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

Question 1.
Is it possible to have a triangle with the following sides?

  1. 2 cm, 3 cm, 5 cm
  2. 3 cm, 6 cm, 7 cm
  3. 6 cm, 3 cm, 2 cm.

Solution:

  1. Since, 2 + 3 > 5
    So the given side lengths cannot form a triangle.
  2. We have, 3 + 6 > 7, 3 + 7 > 6 and 6 + 7 > 3
    i. e., the sum of any two sides is greater than the third side.
    So, these side lengths form a triangle.
  3. We have, 6 + 3 > 2, 3 + 2 \(\ngtr \) 6
    So, the given side lengths cannot form a triangle.

Question 2.
Take any point O in the interior of a triangle PQR. Is

  1. OP + OQ > PQ ?
  2. OQ + OR > QR?
  3. OR + OP > RP ?

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 1

Solution:

  1. Yes ! OP + OQ > PQ …(1)
    Sum of the lengths of any two sides of a triangle is greater than the length of the third side
  2. Yes! OQ + OR > QR …(2)
    Sum of the lengths of any two sides of a triangle is greater than the length of the third side
  3. Yes! OR + OP > RP …(3)
    Sum of the lengths of any two sides of a triangle is greater than the length of the third side

Question 3.
AM is a median of a triangle ABC. Is AB + BC + CA > 2 AM?
(Consider the sides of triangles ∆ ABM and ∆ AMC.)
NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 2
Solution:
Using triangle inequality property in triangles ABM and AMC, we have
AB + BM > AM …(1) and, AC + MC > AM …(2)
Adding (1) and (2) on both sides, we get
AB + (BM + MC) + AC > AM + AM
⇒ AB + BC + AC > 2AM

Question 4.
ABCD is a quadrilateral.
Is AB + BC + CD + DA > AC + BD ?
NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 3
Solution:
In ∆ ABC, AB + BC > AC …(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ ACD, CD + DA > AC …(2)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (1) and (2),
AB + BC + CD + DA > 2AC …(3)
In ∆ ABD, AB + DA > BD …(4)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ BCD, BC + CD > BD …(5)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (4) and (5),
AB + BC + CD + DA > 2BD …(6)
Adding (3) and (6),
2 [AB + BC + CD + DA] > 2 (AC + BD)
⇒ AB + BC + CD + DA > AC + BD.

Question 5.
ABCD is a quadrilateral. Is AB + BC + CD + DA < 2 (AC + BD)?
NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 4
Solution:
In ∆ OAB, OA + OB > AB ….(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In ∆ OBC, OB + OC > BC ….(2)
Sum of the lengths of any two sides of a triangle la greater than the length of the third side.
In ∆ OCA,OC + OA > CA ….(3)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ OAD, OA + OD > AD ….(4)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (1), (2), (3) and (4),
2(OA + OB + OC + OD) > AB + BC + CD + DA
⇒ 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).

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.
∴ Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
∴ We should have
NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 5
∴ The length of the third side should be any length between 3 cm and 27 cm.

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