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. `sqrt2`

Answer

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

### Step 1: Understand the Geometry
The base of the triangle is a line represented by the equation \(x + y = 2\). The vertex is given as \((2, -1)\). We need to find the length of the sides of the triangle.

### 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 \((2, -1)\) to the line \(x + y - 2 = 0\).

The formula for the distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by:

\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Here, \(A = 1\), \(B = 1\), \(C = -2\), and the point is \((x_1, y_1) = (2, -1)\).

### Step 3: Substitute the Values into the Distance Formula
Substituting the 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 4: Relate the Perpendicular Distance to the Side Length
In an equilateral triangle, the relationship between the side length \(s\) and the height \(h\) (which is the perpendicular distance from the vertex to the base) is given by:

\[ h = \frac{s \sqrt{3}}{2} \]

### Step 5: Solve for the Side Length \(s\)
We know that \(h = \frac{1}{\sqrt{2}}\). Setting this equal to the height formula gives:

\[ \frac{1}{\sqrt{2}} = \frac{s \sqrt{3}}{2} \]

Multiplying both sides by 2:

\[ \frac{2}{\sqrt{2}} = s \sqrt{3} \]

Simplifying \(\frac{2}{\sqrt{2}} = \sqrt{2}\):

\[ \sqrt{2} = s \sqrt{3} \]

### Step 6: Isolate \(s\)
Now, isolate \(s\):

\[ s = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{3} = \frac{\sqrt{6}}{3} \]

### Conclusion
Thus, the length of the side of the equilateral triangle is:

\[ \frac{\sqrt{6}}{3} \]