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In the figure above, triangle ABC is an isosceles triangle with AB = BC and BD is the perpendicular dropped from vertex B to the side AC. Point E is marked on BD such that ∠DAE = 450 and ∠ECB = 150. If the ratio of length CD: BD = 1: √3. What is the ratio of the area of triangle BEC to the area of the triangle ABC?
  • a)
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In the figure above, triangle ABC is an isosceles triangle with AB = B...
Given:
  • Isosceles ΔABC
    • AB = BC
    • BD perpendicular to AC
      • Since this is an isosceles Δ, BD will also bisect AC
        • So, AD = CD
 
  • Let’s depict the given and deduced information in the diagram:
  • Equal sides AB and BC are depicted in orange
  • Equal sides AD, ED and DC are depicted in green
We have considered: CD = x
  • Since CD = ED = AD
    • We can denote each of them with “x”
  • Also, BD = BE + ED
    • BE = BD – ED
    • BE = √3x – x = (√3 – 1) x
  • Also AC = AD + CD = x + x = 2x
  • Substituting the value of AC, BE, BD and CD in equation (i) and (ii) we get
    • Area of ΔBEC = ½ * BE * CD = ½ * (√3 – 1) x * x
    • Area of ΔABC = ½ * 2x * √3x
  • Thus the ratio of (Area of ΔBEC)/(Area of ΔABC) is equal to
Therefore, the correct answer is Option A.
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In the figure above, triangle ABC is an isosceles triangle with AB = BC and BD is the perpendicular dropped from vertex B to the side AC. Point E is marked on BD such that ∠DAE = 450and∠ECB = 150. If the ratio of length CD: BD = 1: √3.What is the ratio of the area of triangle BECto the area of the triangle ABC?a)b)c)d)e)Correct answer is option 'A'. Can you explain this answer?
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