Question

The equation of the base of an equilateral triangle is $x+y=2$and the vertex is $(2,-1)$. The length of the side of the triangle is

• A. $\sqrt{\frac{3}{2}}$
• B. $\sqrt{2}$
• C. $\sqrt{\frac{2}{3}}$
• D. None of these

Answer 1

The correct option is C

$\sqrt{\frac{2}{3}}$

Explanation for the correct option:

Finding the length of the side of the triangle:

Step 1: Illustrating the figure using the given information

Considering $C(2,-1)$ as vertex of the equilateral triangle.

The given equation of the base of the equilateral triangle is $x+y=2$

The perpendicular distance from the point $(x_1,y_1)$to a line $ax+by+c=0$ is given by

$d=\left|\frac{(a{x}_1+b{y}_1+c_1)}{\sqrt{(a^2+b^2)}}\right|$

Let's consider the perpendicular distance to side $AB$ be $h$

Therefore, the perpendicular distance from $C(2,-1)$is given by

$$\begin{gathered} h=\left|\frac{(2\times 1–1\times 1–2)}{\sqrt{(1^2+1^2)}}\right| \\ h=\frac{1}{\sqrt{2}}..\dots \left(i\right) \end{gathered}$$

We know that the height for an equilateral triangle is,

$h=\left(\frac{\sqrt{3}}{2}\right)a..\dots .\left(ii\right),$ where $a$ is the side and $h$ is the altitude.

Step 2:Equating the equations $\left(i\right)\&\left(ii\right)$

we get

$$\begin{gathered} \Rightarrow \frac{1}{\sqrt{2}}=\frac{\sqrt{3}a}{2} \\ \Rightarrow a=\sqrt{\frac{2}{3}} \end{gathered}$$

Hence, option (C) is the correct answer.