The equation of the base BC of an equilateral `Delta`ABC is x + y = 2 and A is (2, - 1). The length of the side of the triangle is

Options

A. `sqrt2`
B. `(3/2)^(1//2)`
C. `(1/2)^(1//2)`
D. `(2/3)^(1//2)`

Answer

To find the length of the side of the equilateral triangle ABC, we will follow these steps:

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

### Step 2: Find the perpendicular distance from point A to the line BC
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( x + y - 2 = 0 \), we have:
- \( A = 1 \)
- \( B = 1 \)
- \( C = -2 \)

Substituting the coordinates of point A \( (2, -1) \):
- \( x_1 = 2 \)
- \( y_1 = -1 \)

Now, substituting these values into the distance formula:
\[ d = \frac{|1(2) + 1(-1) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2 - 1 - 2|}{\sqrt{1 + 1}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Use the properties of the equilateral triangle
In an equilateral triangle, the height can be related to the side length \( s \) using the formula:
\[ \text{Height} = \frac{\sqrt{3}}{2} s \]
Here, the height corresponds to the perpendicular distance we calculated.

### Step 4: Set the height equal to the distance and solve for the side length
Setting the height equal to the perpendicular distance:
\[ \frac{\sqrt{3}}{2} s = \frac{1}{\sqrt{2}} \]

Now, solving for \( s \):
\[ s = \frac{1}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} = \frac{2}{\sqrt{6}} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3} \]

### Step 5: Conclusion
The length of the side of the triangle ABC is:
\[ s = \frac{\sqrt{6}}{3} \]