In a triangle ABC , right-angled at A , the altitude through A and the internal bisector of A have lengths 3 and 4, respectively. Find the length of the median through A .

Step-by-step solution

aligned & CAE =45^ = BAE & AD =3 aligned Let B C=a, C A=b, A B=c aligned & 1 2 b c= 1 2 a 3 b c=3 a & 2 b c b+c A 2 =4 6 a b+c 1 2 =4 & 2 2 (b+c)=3 a & 8 (b^2+c^2+2 b c )=9 a^2 & 8 (a^2+6 a )=9 a^2 & 48 a=a^2 a=48 aligned So AF = a 2 =24