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

Options

A. `(4)/(sqrt15)`
B. `(2)/(sqrt15)`
C. `(4)/(3 sqrt3)`
D. `(1)/(sqrt5)`

Answer

To find the length of the sides of the equilateral triangle given the vertex and the equation of its base, we can follow these steps:

### Step 1: Identify the given information
- The vertex of the equilateral triangle is \( A(2, -1) \).
- The equation of the base \( BC \) is given as \( x + 2y = 1 \).

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

### Step 3: Use the formula for the perpendicular distance from a point to a line
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( A = 1 \), \( B = 2 \), \( C = -1 \), \( x_1 = 2 \), and \( y_1 = -1 \):
\[ d = \frac{|1(2) + 2(-1) - 1|}{\sqrt{1^2 + 2^2}} = \frac{|2 - 2 - 1|}{\sqrt{1 + 4}} = \frac{|-1|}{\sqrt{5}} = \frac{1}{\sqrt{5}} \]

### Step 4: Calculate the length of the side of the equilateral triangle
In an equilateral triangle, the height \( h \) from the vertex to the base can be related to the side length \( s \) using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
We have found that the height \( h \) is equal to \( \frac{1}{\sqrt{5}} \). Therefore, we can set up the equation:
\[ \frac{1}{\sqrt{5}} = \frac{\sqrt{3}}{2} s \]

### Step 5: Solve for \( s \)
To find \( s \), we rearrange the equation:
\[ s = \frac{2}{\sqrt{3}} \cdot \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{15}} \]

### Step 6: Simplify the length of the side
To express \( s \) in a more standard form, we can rationalize the denominator:
\[ s = \frac{2\sqrt{15}}{15} \]

Thus, the length of each side of the equilateral triangle is \( \frac{2\sqrt{15}}{15} \).

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