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JEE Advanced2006MathematicsProperties of TrianglesActual

Internal bisector of A of A B C meets side B C at D . A line drawn through D perpendicular to A D intersects the side A C at E and side A B at F . If a, b and c represent sides of A B C , then

Options

  1. AA E is HM of b and c
  2. BA D= 2 b c b+c A 2
  3. CE F= 4 b c b+c A 2
  4. Dthe A E F is isosceles

Correct answer

A. A E is HM of b and c

Step-by-step solution

We have, A B C= A B D+ A C D aligned & 1 2 b c A= 1 2 c A D A 2 + 1 2 b A D A 2 & A D= 2 b c b+c A 2 aligned Again, A E=A D A 2 = 2 b c b+c A E is HM of b and c . aligned E F & =E D+D F=2 D E=2 A D A 2 & =2 2 b c b+c A 2 A 2 = 4 b c b+c A 2 aligned As A D E F and D E=D F and A D is bisector. A E F is isosceles.

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