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If the equation of the base of an equilateral triangle is $x+y$ $=2$ and the vertex is $(2,-1)$, then find the length of the side of the triangle.
Solution:
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Verified Answer
Let $\triangle \mathrm{ABC}$ be an equilateral triangle with side length $(l)$ and perpendicular from $\mathrm{A}$ to $\mathrm{BC}$ as $p$.
$$
\therefore \mathrm{P}=\left|\sin 60^{\circ}=\right| \frac{\sqrt{3}}{2}
$$
Also, $p=\frac{|2-1-2|}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt{2}}$
From (i) and (ii), we get
$$
l \frac{\sqrt{3}}{2}=\frac{l}{\sqrt{2}} \quad \Rightarrow l=\sqrt{\frac{2}{3}}
$$
$$
\therefore \mathrm{P}=\left|\sin 60^{\circ}=\right| \frac{\sqrt{3}}{2}
$$
Also, $p=\frac{|2-1-2|}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt{2}}$
From (i) and (ii), we get
$$
l \frac{\sqrt{3}}{2}=\frac{l}{\sqrt{2}} \quad \Rightarrow l=\sqrt{\frac{2}{3}}
$$
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