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KVPY2020MathematicsProperties of Triangles

Let a = B C ,   b = C A ,   c = A B be the side lengths of a triangle A B C , and m be the length of the median through A . If a = 8 ,   b - c = 2 ,   m = 6 then the nearest integer to b is

Options

  1. A7
  2. B8
  3. C9
  4. D10

Correct answer

B. 8

Step-by-step solution

m 2 = 2   b 2 + 2 c 2 - a 2 4 ⇒ 144 + 64 = 2   b 2 + ( b - 2 ) 2 ⇒ 104 = 2 b 2 - 4 b + 4 ⇒ b 2 - 2 b - 50 = 0 ⇒ ( b - 1 ) 2 = 51 ⇒ b = 1 + 51 ∈ ( 8 , 9 )

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