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

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

NCERT TEXTBOOK EXERCISES

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 > 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 ?

Solution:

1. Yes ! OP + OQ > PQ â€¦(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 â€¦(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 â€¦(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 âˆ† ABM and âˆ† AMC.)

Solution:

Using triangle inequality property in triangles ABM and AMC, we have
AB + BM > AM â€¦(1) and, AC + MC > AM â€¦(2)
Adding (1) and (2) on both sides, we get
AB + (BM + MC) + AC > AM + AM
â‡’ AB + BC + AC > 2AM

Question 4.

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

Solution:

In âˆ† ABC, AB + BC > AC â€¦(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In âˆ† ACD, CD + DA > AC â€¦(2)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
AB + BC + CD + DA > 2AC â€¦(3)
In âˆ† ABD, AB + DA > BD â€¦(4)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In âˆ† BCD, BC + CD > BD â€¦(5)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
AB + BC + CD + DA > 2BD â€¦(6)
2 [AB + BC + CD + DA] > 2 (AC + BD)
â‡’ AB + BC + CD + DA > AC + BD.

Question 5. ABCD is a quadrilateral. Is AB + BC + CD + DA < 2 (AC + BD)?

Solution:

In âˆ† OAB, OA + OB > AB â€¦.(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In âˆ† OBC, OB + OC > BC â€¦.(2)
Sum of the lengths of any two sides of a triangle la greater than the length of the third side.
In âˆ† OCA,OC + OA > CA â€¦.(3)
Sum of the lengths of any two sides of a triangle is greater than 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
Adding (1), (2), (3) and (4),
2(OA + OB + OC + OD) > AB + BC + CD + DA
â‡’ AB + BC + CD + DA < 2 (OA + OB + OC + OD)
â‡’ AB + BC + CD + DA < 2(OA + OC + OB + OD)
â‡’ 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.
âˆ´ Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
âˆ´ We should have

âˆ´ The length of the third side should be any length between 3 cm and 27 cm.

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

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