Question

Using the information given in the following figure, the measure of AE is:

Options

• 20 cm

• 10 cm

• \[6 \frac{2}{3} \mathrm{~cm}\]

• \[20 \sqrt{3} \mathrm{~cm}\]

Answer 1

\[\boldsymbol{6 \frac{2}{3} \mathrm{~cm}}\]

Explanation:

We know that,

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

From figure,

In \[\triangle ABC,\]

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

\[\Rightarrow \sqrt{3} = \frac{20}{BC}\]

\[\Rightarrow BC = \frac{20}{\sqrt{3}}\ \text{cm}.\]

In rectangle BCDE,

Opposite sides of rectangle are equal.

\[\therefore DE = BC = \frac{20}{\sqrt{3}}\ \text{cm}\]

From figure,

In \[\triangle AED,\]

\[\Rightarrow \tan 30^\circ = \frac{AE}{DE}\]

\[\Rightarrow \frac{1}{\sqrt{3}} = \frac{AE}{\frac{20}{\sqrt{3}}}\]

\[\Rightarrow AE = \frac{1}{\sqrt{3}} \times \frac{20}{\sqrt{3}} = \frac{20}{3} = 6\frac{2}{3}\ \text{cm}.\]