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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

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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

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

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