In the figures given below, find AB : (i) (ii) (iii)

(i) In Δ BCD, tan 45° = ⇒ 1 = ⇒ 1 = ⇒ BC = 80 m In Δ ADC, tan 30° = ⇒ ⇒ ⇒ AC = 80 m Now, AB = AC - BC = 80 - 80 m = 80( - 1) m = 80(1.732 - 1) m = 80 x 0.73 m = 58.56 m Hence, the value of AB = 58.56 m. (ii) Given: CD = AC = 40 m In Δ ABC, sin 60° = ⇒ ⇒ ⇒ AB = 40 m = 20 m = 34.64 m Hence, the value of AB = 34.64 m. (iii) Given: AB = BD ⇒ ∠ ADB = ∠ DAB (angles corresponding to the equals sides are always equal) In triangle ABD, sum of all angles of triangle is 180°. ⇒ ∠ ADB + ∠ DAB + ∠ ABD = 180° ⇒ 30° + 30° + ∠ ABD = 180° ⇒ 60° + ∠ ABD = 180° ⇒ ∠ ABD = 180° - 60° = 120° And, ∠ ABD = ∠ BDC + ∠ BCD (exterior angle property) ⇒ 120° = ∠ BDC + 90° ⇒ ∠ BDC = 120° - 90° = 30° In Δ DBC, cos 30° = ⇒ ⇒ ⇒ BD = ⇒ BD = = 23.1m From the figure, BD = AB = 23.1 m Hence, the value of AB = 23.1 m.