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P is a point inside the Δ A B C . Three straight lines parallel to A B , B C and A C are drawn which divides the triangle into six regions. If the area of three triangles formed in these regions are 9,16 and 25 square units, then the area (in sq. units) of Δ A B C is

Options

  1. A81
  2. B256
  3. C144
  4. D64

Correct answer

C. 144

Step-by-step solution

∵ Δ A B C , Δ T U P , Δ P D E , Δ S P R all are similar ∴ ratio of areas = ratio of sides 2 Let, P T = α = A S ∴ S R P T = 4 5 ⇒ S R = 4 α 5 P E P T = 3 5 ⇒ P E = 3 α 5 ⇒ A C = A S + S R + R C = α + 4 α 5 + 3 α 5 = 12 α 5 ∴ a r Δ A B C a r Δ P D E = A C 2 P E 2 = 1 2 2 · α 2 5 2 · 3 2 α 2 5 2 Hence, a r Δ A B C = 16 a r Δ P D E = 144 s q . u n i ts

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