If vertex and base of an equilateral triangle be respectively (2, -1) and $x+y=2$ then length of the side of this equilateral triangle will be :

Question

Accepted Answer

To determine the length of the side of the equilateral triangle, we need to analyze the given information: the vertex of the triangle is at (2, -1), and the base lies on the line $x + y = 2$.

1. **Understanding the Geometry**: 
   - In an equilateral triangle, all sides are equal, and the altitude from any vertex to the opposite side bisects the side and is perpendicular to it.
   - The vertex given is (2, -1), and the base lies on the line $x + y = 2$. The altitude from the vertex (2, -1) to the base will be perpendicular to the line $x + y = 2$.

2. **Finding the Perpendicular Distance**:
   - The line $x + y = 2$ can be rewritten in the form $ax + by + c = 0$ as $x + y - 2 = 0$.
   - The formula for the perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $ax + by + c = 0$ is:
     $$
     d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}
     $$
   - Substituting the values, where $a = 1$, $b = 1$, $c = -2$, and the point $(x_1, y_1) = (2, -1)$:
     $$
     d = \frac{|1 \cdot 2 + 1 \cdot (-1) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2 - 1 - 2|}{\sqrt{2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}}
     $$

3. **Relating Altitude to Side Length**:
   - In an equilateral triangle, the altitude $h$ is related to the side length $s$ by the formula:
     $$
     h = \frac{\sqrt{3}}{2} \cdot s
     $$
   - We have found the altitude $h = \frac{1}{\sqrt{2}}$. Setting this equal to the expression for the altitude in terms of the side length:
     $$
     \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{2} \cdot s
     $$
   - Solving for $s$:
     $$
     s = \frac{1}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} = \frac{2}{\sqrt{6}} = \sqrt{\frac{2}{3}}
     $$

4. **Conclusion**:
   - The calculated side length $s = \sqrt{\frac{2}{3}}$ matches Option 1, confirming it as the correct answer.

5. **Why Other Options are Incorrect**:
   - **Option 2: $\sqrt{\frac{3}{2}}$**, **Option 3: $\sqrt{\frac{1}{2}}$**, and **Option 4: $\sqrt{\frac{5}{6}}$** do not match the derived side length $\sqrt{\frac{2}{3}}$. These values do not satisfy the geometric relationship between the altitude and side length of an equilateral triangle given the specific conditions of this problem.

Thus, Option 1 is the correct answer, as it accurately reflects the calculated side length of the equilateral triangle based on the given conditions.