Heights and Distances Practice Test 1

Let a be the length of a side of square plot ABCD and h, the height of the pole standing at D. Since elevations of P from A or C is 30° and that from B is θ,
∴ In ΔPCD, tan 30° = $\frac{\mathit{h}}{\mathit{a}}$
i.e., $\frac{\mathit{h}}{\mathit{a}}=\frac{1}{\sqrt{3}}$
And in ΔPBD,
$\tan \mathit{\theta }=\frac{\mathit{P}\mathit{D}}{\mathit{B}\mathit{D}}=\frac{\mathit{h}}{\left(\mathit{a}\sqrt{2}\right)}=\frac{1}{\sqrt{6}}$
$\therefore \mathit{B}\mathit{D}=\sqrt{{\left(\mathit{A}{\mathit{B}}^2+\mathit{A}\mathit{D}\right)}^2}=\mathit{a}\sqrt{2}$.