O is a point in the exterior of △ABC. What symbol ‘>’,’<’ or

O is a point in the exterior of △ABC. What symbol ‘>’’ ½ (AB + BC +CA)

FullFormDefinition · 2025-09-14T19:55:37.000000Z

O is a point in the exterior of △ABC. What symbol ‘>’’ ½ (AB + BC +CA)

1.	O is a point in the exterior of △ABC. What symbol ‘>’,’<’ or ‘=’ will you see to complete the statement OA+OB….AB? Write two other similar statements and show that OA + OB + OC > ½ (AB + BC +CA)
Answer» We know that the sum of any two sides of a triangle is always greater than the third side, in △OAB, we have, OA + OB > AB ….. (i) OB + OC > BC …… (ii) OA + OC > CA ….. (iii) On adding equations (i), (ii) and (iii) we get: OA + OB + OB + OC + OA + OC > AB + BC + CA 2(OA + OB + OC) > AB + BC + CA OA + OB + OC > (AB + BC + CA)/2 Or OA + OB + OC > ½ (AB + BC +CA) Hence the proof.

Answer»

We know that the sum of any two sides of a triangle is always greater than the third side, in △OAB, we have,

OA + OB > AB ….. (i)

OB + OC > BC …… (ii)

OA + OC > CA ….. (iii)

On adding equations (i), (ii) and (iii) we get:

OA + OB + OB + OC + OA + OC > AB + BC + CA

2(OA + OB + OC) > AB + BC + CA

OA + OB + OC > (AB + BC + CA)/2

OA + OB + OC > ½ (AB + BC +CA)

Hence the proof.