Question

In the given figure, $D$ is the mid-point of $BC$, then the value of $\frac{\cot \left(y°\right)}{\cot \left(x°\right)}$ is

• A. $2$
• B. $\frac{1}{4}$
• C. $\frac{1}{3}$
• D. $\frac{1}{2}$

Answer 1

The correct option is D

$\frac{1}{2}$

Explanation for the correct option:

Given: $D$ is the mid-point of $BC$
$\Rightarrow BD=CD$

$\cot \left(x°\right)=\frac{AC}{CD}$

$$\begin{gathered} \cot \left(y°\right)=\frac{AC}{BC} \\ \cot \left(y°\right)=\frac{AC}{BD+CD} \\ \cot \left(y°\right)=\frac{AC}{CD+CD}\left[\because BD=CD\right] \\ \cot \left(y°\right)=\frac{AC}{2\times CD} \end{gathered}$$

$$\begin{gathered} \frac{\cot \left(y°\right)}{\cot \left(x°\right)}=\frac{{\displaystyle \frac{AC}{2\times CD}}}{{\displaystyle \frac{AC}{CD}}} \\ \frac{\cot \left(y°\right)}{\cot \left(x°\right)}=\frac{1}{2} \end{gathered}$$

Therefore, $\frac{\cot \left(y°\right)}{\cot \left(x°\right)}=\frac{1}{2}$

Hence, option (D) is the correct option.