Find AD.

(i)

(ii)

(i) From figure,

BE = CD = 20 m and DE = CB = 5 m.

In △ABE,

$\text{tan 32°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow 0.6249 = \dfrac{AE}{BE} \\[1em] \Rightarrow AE = 0.6249 \times BE \\[1em] \Rightarrow AE = 0.6249 \times 20 \\[1em] \Rightarrow AE = 12.498 \text{ m}.$

AD = AE + DE = 12.498 + 5 = 17.498 ≈ 17.5 m.

Hence, AD = 17.5 meters.

(ii) We know that,

An exterior angle is equal to the sum of two opposite interior angles.

∴ ∠ACD = ∠ABC + ∠BAC

Also, ∠ABC = ∠BAC (As, angles opposite to equal sides are equal)

∴ ∠ACD = 2∠ABC

⇒ 2∠ABC = 48°

⇒ ∠ABC = 24°.

In △ABD,

$\text{sin 24°} = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}} \\[1em] \Rightarrow 0.4067 = \dfrac{AD}{AB} \\[1em] \Rightarrow AD = 0.4067 \times AB \\[1em] \Rightarrow AD = 0.4067 \times 30 \\[1em] \Rightarrow AD = 12.20 \text{ m}.$

Hence, AD = 12.20 meters.