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KVPY2020MathematicsProperties of Triangles

In a triangle A B C , a point D is chosen on B C such that B D : D C = 2 : 5 . Let P be a point on the circumcircle A B C such that ∠ P D B = ∠ B A C . Then P D : P C is :-

Options

  1. A2 : 5
  2. B2 : 5
  3. C2 : 7
  4. D2 : 7

Correct answer

D. 2 : 7

Step-by-step solution

B D D C = 2 5 ∠ P D B = ∠ B A C = θ let ∠ P C D = α ⇒ ∠ D P C = θ - α ∠ B A C = ∠ B P C = θ (angle in the same segment) ⇒ B P D = θ - ( θ - α ) = α so Δ P C B ~ Δ P D B P C D P = B C P B = P B B D P C D P 2 = B C P B × P B B D = B C B D P C D P = B C B D = 7 λ 2 λ = 7 2 D P P C = 2 7

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