Question

The length of DC is ______.

Options

• \[15 \sqrt{3} \mathrm{~cm}\]

• \[15(\sqrt{3}-1) \mathrm{cm}\]

• \[15(\sqrt{3}+1) \mathrm{cm}\]

• \[10 \sqrt{3} \mathrm{~cm}\]

Answer 1

The length of DC is \[\underline{\boldsymbol 10 \sqrt{3} \mathrm{~cm}}\].

Explanation:

We know that,

\[\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}\]

From figure,

In \[\triangle ABC,\]

\[\Rightarrow \tan 60^\circ = \frac{AB}{BC}\]

\[\Rightarrow \sqrt{3} = \frac{15}{BC}\]

\[\Rightarrow BC = \frac{15}{\sqrt{3}}\]

Rationalizing the denominator,

\[\Rightarrow BC = \frac{15}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\]

\[\Rightarrow BC = \frac{15\sqrt{3}}{3}\]

\[\Rightarrow BC = 5\sqrt{3}.\]

In \[\triangle ABD,\]

\[\Rightarrow \tan 30^\circ = \frac{AB}{BD}\]

\[\Rightarrow \frac{1}{\sqrt{3}} = \frac{15}{BD}\]

\[\Rightarrow BD = 15\sqrt{3}\ \text{cm}.\]

From figure,

\[DC = BD - BC = 15\sqrt{3} - 5\sqrt{3} = 10\sqrt{3}\ \text{cm}.\]