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KVPY2020MathematicsProperties of Triangles

In a triangle A B C , the altitude A D and the median A E divide ∠ A into three equal parts. If B C = 28 , then the nearest integer to A B + A C is

Options

  1. A38
  2. B37
  3. C36
  4. D33

Correct answer

A. 38

Step-by-step solution

Δ A B E is isosceles ⇒ B D = D E = 7 Δ A D C : tan 90 - 2 θ = A D 21 . . . 1 Δ A D E : tan 90 - θ = A B 7 . . . 2 Divide tan θ tan 2 θ = 1 3 ⇒ 1 - tan 2 θ 2 = 1 3 1 - tan 2 θ = 2 3 ⇒ tan θ = 1 3 ⇒ θ = 30 ° Δ A B D : cos 90 - θ = B D C = sin θ C = 7 cosec θ = 14 ∆ A D C : cos 90 - 2 θ = D C b = sin 2 θ b = 42 3 = 14 3 [ b + c ] = 38

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