Question

$ABCD$ is a rectangular with $AD=12\text{cm}$ and $DC=20\text{cm}.$ Line segment $DE$ is drawn making an angle of ${30}^{\circ }$ with $AD,$ intersecting $AB$ in $E.$ Find the length of $DE$ and $AE.$

Answer 1

Given $\Rightarrow$ $AD=12\text{cm}$

$DC=20\text{cm}$

In $△AED,$

$\begin{array}{cc}\tan {30}^{\circ }& =\frac{AE}{AD}\\ & \\ \frac{1}{\sqrt{3}}& =\frac{AE}{12}\\ & \\ AE& =\frac{12}{\sqrt{3}}\\ & \\ & =4\sqrt{3}\text{cm}\end{array}$

And,

$\begin{array}{cc}\sin {30}^{\circ }& =\frac{AE}{ED}\\ & \\ \frac{1}{2}& =\frac{4\sqrt{3}}{ED}\\ & \\ ED& =8\sqrt{3}\text{cm}\end{array}$

Hence, $DE$ and $AE$ is equal to $8\sqrt{3}\text{cm}$ and $4\sqrt{3}\text{cm}$ respectively.