Question

The equation of the base $BC$ of an equilateral $∆ABC$ is $x+y=2$ and $A$ is $\left(2,-1\right)$. The length of the side of the triangle is

• A. $\sqrt{2}$
• B. ${\left(\frac{3}{2}\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{1}{2}\right)}^{\frac{1}{2}}$
• D. ${\left(\frac{2}{3}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is D

${\left(\frac{2}{3}\right)}^{\frac{1}{2}}$

Explanation for the correct option:

Finding the length of side:

Equation of the base is $x+y=2$.

$∆ABC$ is an equilateral triangle,$M$ is midpoint of side $BC$ and $AM\perp BC$.

$$\begin{gathered} AM=\left|\frac{x+y-2}{\sqrt{1+1}}\right| \\ =\left|\frac{2-1-2}{\sqrt{2}}\right| \\ =\frac{1}{\sqrt{2}} \end{gathered}$$

Now, $\tan \left(\angle ABM\right)=\frac{AM}{BM}$

Since, $∆ABC$ is an equilateral triangle, $\angle ABM=60°$.

$$\begin{gathered} \Rightarrow \tan \left(60°\right)=\frac{1/\sqrt{2}}{BM} \\ \Rightarrow \sqrt{3}=\frac{1/\sqrt{2}}{BM} \\ \Rightarrow BM=\frac{1}{\sqrt{2}\times \sqrt{3}} \\ \Rightarrow BM=\frac{1}{\sqrt{6}} \end{gathered}$$

$M$is midpoint of side $BC$ .

$$\begin{gathered} \therefore BC=2\times BM \\ =2\times \frac{1}{\sqrt{6}} \\ =\frac{\sqrt{2}}{\sqrt{3}} \\ ={\left(\frac{2}{3}\right)}^{\frac{1}{2}} \end{gathered}$$

Therefore, the correct answer is option (D).