Find the length of AD.

Given :

∠ABC = 60°,
∠DBC = 45°
and BC = 40 cm.

In Δ BDC,

$\text{tan 45°} = \dfrac{Perpendicular}{Base}\\[1em] ⇒ 1 = \dfrac{DC}{BC}\\[1em] ⇒ 1 = \dfrac{DC}{40}\\[1em] ⇒ DC = 40$

In Δ ABC,

$\text{tan 60°} = \dfrac{Perpendicular}{Base}\\[1em] ⇒ \sqrt3 = \dfrac{AC}{BC}\\[1em] ⇒ \sqrt3 = \dfrac{AC}{40}\\[1em] ⇒ AC = 40\sqrt3 = 69.28$

AD = AC - DC

⇒ AD = 69.28 - 40 = 29.28 cm

Hence, AD = 29.28 cm.