Question:

Directions: The following question has four choices, out of which one or more are correct.The internal bisector of ∠A of a triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and side AB at F. If a,b and c represent the sides of ΔABC, then

Updated On: Jul 30, 2024
  • (A) AE is HM of b and c.
  • (B) AD=2bcb+ccos⁡A2
  • (C) EF=4bcb+csin⁡A2
  • (D) △AEF is isosceles
Show Solution
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The Correct Option is A, D

Solution and Explanation

Explanation:
 We have ΔABC=ΔABD+ΔACD⟹12bcsinA=12cADsin⁡A2+12b×ADsin⁡A2⟹AD=2bcb+ccos⁡A2 Again AE=ADsec⁡A2=2bcb+c⟹AE is HM of b and c. EF=ED+DF=2DE=2×ADtanA2=2×2bcb+c×cos⁡A2×tan⁡A2=4bcb+csin⁡A2As AD⊥EF and DE=DF and AD is bisector ⟹AEF is isosceles.
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