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.

- The Triangle and its Properties Class 7 Ex 6.1
- The Triangle and its Properties Class 7 Ex 6.2
- The Triangle and its Properties Class 7 Ex 6.3
- The Triangle and its Properties Class 7 Ex 6.5
- The Triangle and its Properties Class 7 MCQ

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?

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

**Solution:**

- Since, 2 + 3 > 5

So the given side lengths cannot form a triangle. - 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. - 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

- OP + OQ > PQ ?
- OQ + OR > QR?
- OR + OP > RP ?

**Solution:**

- 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 - 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 - 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.)

**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 ?

**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)?

**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

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

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