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KVPY2018MathematicsProperties of Triangles

Let a ,   b ,   c be the side-length of a triangle and l ,   m ,   n be the lengths of its medians. Put K = l + m + n a + b + c Then, as a ,   b ,   c vary, K can assume every value in the interval

Options

  1. A1 4 , 2 3
  2. B1 2 , 4 5
  3. C3 4 , 1
  4. D4 5 , 5 4

Correct answer

C. 3 4 , 1

Step-by-step solution

Let Δ A B C B C = a A C = b A B = c and median of Δ A B C A D = l B E = m C F = n A D is median, ∴ A D A B + B C 2 ∴ l b + c 2 Similarly, m a + b 2 and n a + c 2 ∴ l + m + n a + b + c ⇒ l + m + n a + b + c 1 ... ( i ) Also in Δ B G C , B G + G C > B C ∴ 2 3 ( m + n ) > a Similarly, 2 3 ( n + l ) > b and 2 3 ( m + l ) > c ∵ 4 3 ( l + m + n ) > a + b + c ⇒ l + m + n a + b + c > 3 4 ... ( ii ) From Eqs. ( i ) and ( ii ), we get l + m + n a + b + c ∈ 3 4 , 1

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