The Triangle and its Properties

Question

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

Answer

In a triangle, the sum of the lengths of either two sides is always greater than the third side.

Considering ΔOAB,

OA + OB > AB (i)

In ΔOBC,

OB + OC > BC (ii)

In ΔOCD,

OC + OD > CD (iii)

In ΔODA,

OD + OA > DA (iv)

Adding equations (i), (ii), (iii), and (iv), we obtain

OA + OB + OB + OC + OC + OD + OD + OA > AB + BC + CD + DA

2OA + 2OB + 2OC + 2OD > AB + BC + CD + DA

2OA + 2OC + 2OB + 2OD > AB + BC + CD + DA

2(OA + OC) + 2(OB + OD) > AB + BC + CD + DA

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

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

Yes, the given expression is true.

Sponsor Area

Some More Questions From The Triangle and its Properties Chapter

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

Take any point O in the interior of a triangle PQR. Is
(i) OP + OQ > PQ?
(ii) OQ + OR > QR?
(iii) OR + OP > RP?

AM is a median of a triangle ABC.
Is AB + BC + CA > 2 AM?
(Consider the sides of triangles ΔABM and ΔAMC.)

ABCD is quadrilateral.
Is AB + BC + CD + DA >AC + BD?