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

• A. (A) AE is HM of b and c.
• B. (B) AD=2bcb+ccos⁡A2
• C. (C) EF=4bcb+csin⁡A2
• D. (D) △AEF is isosceles

Answer 1

Answer: (A), (D)

(A) AE is HM of b and c.

Answer 2

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.