In the following figure :

OA = 2 cm = AB = 2BC = $\dfrac{1}{2}$ CD = $\dfrac{1}{6}$ DE

Find the length of OE.

Given: OA = AB = 2 cm

2BC = 2 cm ⇒ BC = $\dfrac{2}{2}$ = 1 cm

$\dfrac{1}{2}$ CD = 2 cm ⇒ CD = 2 x 2 = 4 cm

$\dfrac{1}{6}$ DE ⇒ DE = 6 x 2 = 12 cm

In Δ ABO, using Pythagorean theorem,

⇒ BO² = AB² + AO²

= (2)² + (2)²

= 4 + 4

= 8

⇒ BO = $\sqrt{8}$ = 2 $\sqrt{2}$ cm

Similarly, in Δ OBC, using Pythagorean theorem,

⇒ CO² = OB² + BC²

= (2 $\sqrt{2}$)² + 1²

= 8 + 1

= 9

⇒ CO = $\sqrt{9}$ = 3 cm

Similarly, in Δ OCD, using Pythagorean theorem,

⇒ DO² = OC² + CD²

= 3² + 4²

= 9 + 16

= 25

⇒ DO = $\sqrt{25}$ = 5 cm

In Δ ODE, using Pythagorean theorem,

⇒ OE² = OD² + DE²

= 5² + 12²

= 25 + 144

= 169

⇒ OE = $\sqrt{169}$ = 13 cm

Hence, the length of OE = 13cm.