CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If O is a point within triangle ABC, show that,

  1. AB+AC>OB+OC
  2. AB+BC+CA>OA+OB+OC
  3. OA+OB+OC>12(AB+BC+CA)

Open in App
Solution


In ΔABC,

AB+AC>BC ….(i) [ Sum of two sides of a triangle is greater than the third side.]

Consider ΔOBC,

OB+OC>BC …(ii)

Subtracting (i) from (ii) we get,

(AB+AC)(OB+OC)>(BCBC)

(1) AB+AC>OB+OC

From (1), AB+AC>OB+OC

Similarly, AB+BC>OA+OC

and AC+BC>OA+OB

Adding both sides of these three inequalities, we get,

(AB+AC)+(AB+BC)+(AC+BC)>(OB+OC)+(OA+OC)+(OA+OB)

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

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

Again in ΔABC,

OA+OB>AB

OB+OC>BC

OA+OC>AC

On adding the above equations,

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

Hence,

(3) OA+OB+OC>12(AB+BC+CA)


flag
Suggest Corrections
thumbs-up
24
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Axioms
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon