**The Triangle and Its Properties**

**Exercise 6.4**

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, 5 cm

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

the sum of the two sides are

2 + 3 = 5

Third side = 5

the sum of the two sides = third side

therefore, triangle is not possible

(ii) 3 cm, 6 cm, 7 cm

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

the sum of the two sides are

3 + 6 = 9

Third side = 7

the sum of the two sides > the third side

the sum of the two sides are

3 + 7 = 10

Third side = 6

the sum of the two sides > the third side

the sum of the two sides are

7 + 6 = 13

Third side = 3

the sum of the two sides > the third side

therefore, triangle is possible

(iii) 6 cm, 3 cm, 2 cm

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

the sum of the two sides are

2 + 3 = 5

Third side = 6

the sum of the two sides < the third side

therefore, triangle is not possible

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

(i) OP + OQ > PQ?

Yes, in DOPQ

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

(ii) OQ + OR > QR?

Yes, in DOQR

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

(iii) OR + OP > RP?

Yes, in DOPR

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

3). AM is a median of a triangle ABC.

Is AB + BC + CA > 2 AM?

(Consider the sides of triangles DABM and DAMC.)

In DAMB

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AB + BM > AM ……………………(I)

In DAMC

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AC + CM > AM ……………………(II)

Adding I and II

AB + BM + AC + CM > AM + AM

AB + __BM + CM__ + AC > AM + AM

AB + BC + AC > 2AM

4). ABCD is a quadrilateral.

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

In DABC

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AB + BC > AC ……………………(I)

In DBCD

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

BC + CD > BD ……………………(II)

In DCDA

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

CD + DA > AC ……………………(III)

In DDAB

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

DA + AB > BD ……………………(IV)

Adding I, II, III and IV

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

2AB + 2BC + 2CD + 2DA > 2AC + 2BD

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

Dividing both inequalities with 2, we get

AB + BC + CD + DA > AC + BD

5). ABCD is quadrilateral. Is

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

Let P be the point of intersection of diagonals AC and BD

In DAPB

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AP + PB > AB ……………………(I)

Similarly In DPCB

PB + PC > BC ……………………(II)

Similarly In DAPD

AP + PD > AD ……………………(III)

Similarly In DCPD

PD + PC > CD ……………………(IV)

Adding I, II, III and IV

AP + PB + PB + PC + AP + PD + PD + PC > AB + BC + AD + CD

2AP + 2PB + 2PC + 2AP > AB + BC + CD + AD

2AP + 2PC + 2PB + 2 > AB + BC + CD + AD

2AC + 2BD > AB + BC + CD + AD

2(AC + BD) > AB + BC + CD + AD

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

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?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Given sides of the triangle are 12 and 15 cm

Therefore, the third side is less than 12 + 15 = 27 cm

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

15 – 12 = 3

The third side cannot be less than 3 cm

Therefore, the length of the third side falls between 3 cm and 27 cm.

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