wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If a, b, c are the sides of a triangle, then ac+ab+ba+bc+cb+ca3

A
True
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
False
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is A True
Since a, b, c are the sides of the triangle we have a, b, c > 0 and we know that the sum of any two sides of a triangle is greater than the third side.
a+b-c, b+c-a, c+a-b are all positive.
Applying AMGM to the quantities.
a b c
c+a -b , a+b-c, b+c-a
we have ac+ab+ba+bc+cb+ca
33abc(c+ab)(a+bc)(b+ca)
We have
a2a2(bc)2
or a2(a+bc)(ab+c)
Similarly
b2(b+ca)(bc+a)
c2(c+ab))(ca+b))
Multiplying together we get
a2b2c2(a+bc)2(b+ca)2(c+ab)2
Taking he positive root both sides we have
abc(a+bc)(b+ca)(c+ab)
abc(a+bc)(b+ca)(c+ab)1
3abc(a+bc)(b+ca)(c+ab)1
ac+ab+ba+bc+cb+ca3

flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Cosine Rule
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon