If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, -1), then the length of the side of the triangle is (in unit)

Options

A. `sqrt(2/3)`
B. `sqrt(3/2)`
C. `sqrt(1/2)`
D. `sqrt(5/6)`

Answer

To find the length of the side of the equilateral triangle given the base equation and the vertex, we can use the formula for the distance from a point to a line.

### Step-by-Step Solution:

**Step 1: Identify the equation of the base and the vertex.**
- The equation of the base is given as \(x + y = 2\).
- The vertex of the triangle is given as \( (2, -1) \).

**Step 2: Rewrite the equation of the line in the standard form.**
- The equation \(x + y = 2\) can be rewritten in the form \(Ax + By + C = 0\):
\[ x + y - 2 = 0 \]
Here, \(A = 1\), \(B = 1\), and \(C = -2\).

**Step 3: Use the formula for the distance from a point to a line.**
- The formula for the 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 \(A = 1\), \(B = 1\), \(C = -2\), \(x_1 = 2\), and \(y_1 = -1\):
\[ d = \frac{|1(2) + 1(-1) - 2|}{\sqrt{1^2 + 1^2}} \]

**Step 4: Simplify the expression.**
- Calculate the numerator:
\[ |2 - 1 - 2| = | -1 | = 1 \]
- Calculate the denominator:
\[ \sqrt{1^2 + 1^2} = \sqrt{2} \]
- Therefore, the distance \(d\) is:
\[ d = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

**Step 5: Calculate the length of the side of the equilateral triangle.**
- The length of the side \(s\) of the equilateral triangle is given by:
\[ s = 2d \]
- Substituting the value of \(d\):
\[ s = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} \]

Thus, the length of the side of the triangle is \(\sqrt{2}\) units.

### Final Answer:
The length of the side of the triangle is \(\sqrt{2}\) units.