Question

abc is a right triangle and right angled at b ad and ce are 2 medians from a and c respectively if ac=5 cm and ad=3 root 5 cm find the length of ce

Answer 1

See diagram.
AC = 5 cm.  AD = 3 cm..      if AD = 3√5 cm then it is wrong,, as  AD < AC. AC is the longest line segment in the triangle ABC.

let BD = x,  BC = 2 x,    AE = y,  AB = 2 y

AD² = AB² + BD² = 4 y² + x²  = (3)² = 9 cm²      ---- (1)

AC² = AB² + BC² =>  4 y² + 4 x² = 5² = 25 cm²      ---- (2)

   (2) - (1) =>  3 x² = 16        => x = 4/√3 cm
               => y = √(11/12) cm

CE² = BC² + BE² = (2x)² + y² = 4 x² + y²  = 64/3 + 11/12 = 69/4 cm²
   CE = √69/2 cm

Since ABC is a right angle triangle,  BF = BE/2  = CE / 2  = 2.5 cm
  So BO = 2*2.5/3 = 5/3 cm  and  OE = 2.5/3 cm