Question

The equation of the base of an equilateral triangle is \( x + y = 2 \) and its opposite vertex is \( (2,1) \). If \( m_1, m_2 \) are the slopes of the other two sides and the length of its side is \( a \), then \( |m_1 - m_2| + a\sqrt{2} = \):

• A. \( 8\sqrt{5} \)
• B. \( \frac{8}{\sqrt{5}} \)
• C. \( \frac{2\sqrt{2}}{3} \)
• D. \( \frac{8\sqrt{2}}{3} \)

Hint:

For geometry problems involving transformations, always compute perpendicular distances using the formula: \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Answer 1

The Correct Option is D

The base equation is \( x + y = 2 \), and the opposite vertex is \( (2,1) \). The perpendicular distance from the vertex to the base is computed using: \[ d = \frac{|2 + 1 - 2|}{\sqrt{1^2 + 1^2}} = \frac{1}{\sqrt{2}} \] Using equilateral triangle properties: \[ a = \frac{2 \times d}{\sin 60^\circ} = \frac{2 \times \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}} \] Solving for \( |m_1 - m_2| \): \[ |m_1 - m_2| = \frac{4\sqrt{2}}{3} \] Final value: \[ |m_1 - m_2| + a\sqrt{2} = \frac{8\sqrt{2}}{3} \]