Triangle ABC is right angled at B. AD and CE are the two medians drawn from A and C respectively. If AC = 5cm, AD = cm. Find the length of CE.

We have

Given that,

∆ABC is a right-angled triangle at ∠B.

BD = DC

AE = EB

Values:

AC = 5 cm

To find Length of CE.

Take ∆ABC,

Apply Pythagoras theorem in ∆ABC, we get

AC² = AB² + BC²

⇒ (5)² = (2BE)² + BC²

⇒ 25 = 4(BE)² + BC² …(i)

Take ∆ABD,

Apply Pythagoras theorem in ∆ABD, we get

AB² + BD² = AD² …(ii)

Put the values in the above equation, we get

⇒ 45 = 16BE² + BC² …(iii)

For solving equations (i) and (iii), multiply equation (i) with 4.

(i) → 100 = 16BE² + 4BC² …(iv)

Solve equations (iii) and (iv),

Take 25 = 4(BE)² + BC²,

⇒ AC² = 4BE² + BC²

⇒ 4BE² = AC² - BC²

Now, in ∆BEC, applying Pythagoras theorem, we can write

BE² + BC² = CE²

⇒ CE² = 20

⇒ CE = √20

⇒ CE = √(2 × 2 × 5)

⇒ CE = 2√5

Thus, length of CE = 2√5 cm.