Find :

(i) BC

(ii) AD

(iii) AC

(i) In Δ ABC,

$\text{tan 30°} = \dfrac{Perpendicular}{Base}\\[1em] ⇒ \dfrac{1}{\sqrt3} = \dfrac{AB}{BC}\\[1em] ⇒ \dfrac{1}{\sqrt3} = \dfrac{12}{BC}\\[1em] ⇒ BC = 12 \sqrt3\\[1em] ⇒ BC = 20.78$

Hence, BC = 20.78 cm.

(ii) In Δ ABC, according to angle sum property,

∠ ABC + ∠ BAC + ∠ ACB = 180°

⇒ 90° + ∠ BAC + 30° = 180°

⇒ 120° + ∠ BAC = 180°

⇒ ∠ BAC = 180° - 120°

⇒ ∠ BAC = 60° = ∠ BAD

In Δ ABD,

$\text{cos 60°} = \dfrac{Base}{Hypotenuse}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{AD}{AB}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{AD}{12}\\[1em] ⇒ AD = \dfrac{12}{2}\\[1em] ⇒ AD = 6$

Hence, AD = 6 cm.

(iii) In Δ ABC,

$\text{sin 30°} = \dfrac{Perpendicular}{Hypotenuse}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{AB}{AC}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{12}{AC}\\[1em] ⇒ AC = 12 \times 2\\[1em] ⇒ AC = 24$

Hence, AC = 24 cm.