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Question: If a, b, c and d are the sides of a quadrilateral, then the value of \(\frac { a ^ { 2 } + b ^ { 2 ...

If a, b, c and d are the sides of a quadrilateral, then the value of a2+b2+c2d2\frac { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } { d ^ { 2 } } is always greater than

A

1

B

1/2

C

1/3

D

1/4

Answer

1/3

Explanation

Solution

Since (a – b)2 + (b – c)2 + (c – a)2 ≥ 0

⇒ 2(a2 + b2 + c2) ≥ 2ab + 2bc + 2ca

⇒ 3 (a2 + b2 + c2) ≥ a2 + b2 + c2 + 2ab + 2bc + 2ca ⇒ 3 (a2 + b2 + c2) ≥ (a + b + c)2 > d2

⇒ 3 (a2 + b2 + c2) > d2 ⇒ a2+b2+c2d2\frac { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } { d ^ { 2 } }>13\frac { 1 } { 3 }