Q: If O is the midpoint within increment ABC, show that:
i) AB + AC> OB + OC
ii)AB + BC+CA >OA+OB+OC
iii) OA+OB+OC>1 half(AB + BC + CA)

Asked by kumar.ashlesha | 14th Aug, 2014, 07:50: PM

Expert Answer:

(i)
Consider the following figure.
Produce OB to BD to meet AC at D.
WE know that sum of any two sides of a atriangle is always less than the third side of the triangle.
Thus, we have,
T h u s comma space f r o m space t h e space t r i a n g l e space A B C comma A B plus A C greater than B C... left parenthesis 1 right parenthesis S i m i l a r l y space f r o m space t h e space t r i a n g l e space B C D comma space w e space h a v e B D plus C D greater than B C... left parenthesis 2 right parenthesis T h u s comma space f r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis space w e space h a v e A B plus A C greater than B D plus C D... left parenthesis 3 right parenthesis W e space h a v e comma space O B plus O D equals B D... left parenthesis 4 right parenthesis T h u s comma space f r o m space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e A B plus A C greater than O B plus O D plus C D... left parenthesis 5 right parenthesis B u t space O D plus C D greater than O C... left parenthesis 6 right parenthesis S o comma space f r r o m space left parenthesis 5 right parenthesis space a n d space left parenthesis 6 right parenthesis comma space w e space h a v e A B plus A C greater than O B plus O C

Answered by Vimala Ramamurthy | 16th Aug, 2014, 11:47: AM

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