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

Options

A. `sqrt((1/2))`
B. `sqrt((3/2))`
C. `sqrt((2/3))`
D. `sqrt(2)`

Answer

To find the length of the side of the equilateral triangle with the base given by the equation \(x + y = 2\) and the vertex at the point \((2, -1)\), we can follow these steps:

### Step 1: Identify the equation of the base and the vertex
The equation of the base of the triangle is given as:
\[ x + y = 2 \]
The vertex of the triangle is at the point:
\[ A(2, -1) \]

### Step 2: Find the perpendicular distance from the vertex to the base
To find the length of the side of the triangle, we first need to calculate the perpendicular distance from the vertex \(A(2, -1)\) to the line \(x + y - 2 = 0\).

The formula for the perpendicular distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
In our case, \(A = 1\), \(B = 1\), \(C = -2\), and the point is \((x_1, y_1) = (2, -1)\).

Substituting these values into the formula:
\[ d = \frac{|1 \cdot 2 + 1 \cdot (-1) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2 - 1 - 2|}{\sqrt{1 + 1}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Use the area formula for the equilateral triangle
The area \(A\) of an equilateral triangle can also be expressed in terms of the side length \(s\):
\[ A = \frac{\sqrt{3}}{4} s^2 \]
We can also express the area in terms of the height \(h\) (which is the perpendicular distance we just calculated):
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Since the base is the length of the line segment between the two points where the triangle meets the line \(x + y = 2\), we can denote this length as \(s\). Thus:
\[ A = \frac{1}{2} \times s \times \frac{1}{\sqrt{2}} \]

### Step 4: Equate the two area expressions
Setting the two area expressions equal gives:
\[ \frac{\sqrt{3}}{4} s^2 = \frac{1}{2} \times s \times \frac{1}{\sqrt{2}} \]

### Step 5: Solve for \(s\)
Rearranging the equation:
\[ \frac{\sqrt{3}}{4} s^2 = \frac{s}{2\sqrt{2}} \]
Multiplying both sides by \(4\sqrt{2}\):
\[ \sqrt{3} \cdot 2\sqrt{2} s^2 = 4s \]
Dividing both sides by \(s\) (assuming \(s \neq 0\)):
\[ 2\sqrt{6} s = 4 \]
Thus:
\[ s = \frac{4}{2\sqrt{6}} = \frac{2}{\sqrt{6}} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3} \]

### Conclusion
The length of the side of the equilateral triangle is:
\[ s = \frac{\sqrt{6}}{3} \]