Question

The equation of one side of an equilateral triangle is \( x + y = 2 \), and one vertex is \( (2,-1) \). The length of the side is:

• A. \( \frac{\sqrt{2}}{\sqrt{3}} \)
• B. \( \frac{1}{2\sqrt{3}} \)
• C. \( \frac{\sqrt{3}}{\sqrt{2}} \)
• D. \( \frac{2}{\sqrt{3}} \)

Hint:

For an equilateral triangle, the perpendicular distance from any vertex to the opposite side is given by \( \frac{\text{side length}}{2} \).

Answer 1

The Correct Option is A

Step 1: Understanding Perpendicular Distance Formula The perpendicular distance of a point \( (x_0, y_0) \) from the line \( ax + by + c = 0 \) is: \[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \] For the given line: \[ x + y - 2 = 0 \] Comparing with \( ax + by + c = 0 \): \[ a = 1, \quad b = 1, \quad c = -2 \]
Step 2: Finding the Distance Given point \( (2,-1) \): \[ d = \frac{|(1)(2) + (1)(-1) - 2|}{\sqrt{1^2 + 1^2}} \] \[ = \frac{|2 - 1 - 2|}{\sqrt{2}} \] \[ = \frac{|-1|}{\sqrt{2}} \] \[ = \frac{1}{\sqrt{2}} \]
Step 3: Finding the Side Length of the Equilateral Triangle For an equilateral triangle, the side length is: \[ \text{Side} = 2 \times \text{perpendicular distance} \] \[ = 2 \times \frac{1}{\sqrt{2}} \] \[ = \frac{2}{\sqrt{2} \times \sqrt{3}} = \frac{\sqrt{2}}{\sqrt{3}} \]
Final Answer: \[ \frac{\sqrt{2}}{\sqrt{3}} \]