<p>A vertex of an equilateral triangle is \((2, 3)\) and the equation of the opposite side is \(x + y = 2\). Find the equations of the other two sides and the length of each side of the triangle.</p>
Step-by-Step Solution
Key Concept: Use the property that in an equilateral triangle, the angle from a vertex to the opposite side is 60°. The perpendicular distance from vertex to opposite side equals (√3/2) × side length, and the slopes of the two sides through the vertex make 60° angles with each other.
<p><strong>Step 1:</strong> Find perpendicular distance from (2,3) to line x+y=2.</p><p>Distance d = |2+3-2|/√2 = 3/√2 = (3√2)/2</p><p><strong>Step 2:</strong> In an equilateral triangle, if h is the perpendicular distance from vertex to opposite side, then h = (√3/2)a where a is the side length.</p><p>Therefore: (3√2)/2 = (√3/2)a → a = (3√2)/√3 = √6</p><p><strong>Step 3:</strong> The line x+y=2 has slope m₀ = -1. The two sides through (2,3) make angles of ±60° with the perpendicular to x+y=2.</p><p>The perpendicular to x+y=2 has slope 1 (perpendicular to -1).</p><p><strong>Step 4:</strong> Using the angle formula tan(60°) = √3, if the perpendicular has slope 1, the two sides have slopes:</p><p>m = (1+√3)/(1-√3) = (1+√3)²/(1-3) = -(2+2√3)/2 = -(1+√3) = (√3+1)/(√3-1) (after rationalization)</p><p>m = (1-√3)/(1+√3) = (1-√3)²/(1-3) = (4-2√3)/(-2) = (√3-1)/(√3+1) (after rationalization)</p><p><strong>Step 5:</strong> Equations of the two sides through (2,3):</p><p><strong>Side 1:</strong> y - 3 = [(√3+1)/(√3-1)](x - 2)</p><p><strong>Side 2:</strong> y - 3 = [(√3-1)/(√3+1)](x - 2)</p><p>∴ <strong>Answer:</strong> The equations are y - 3 = [(√3+1)/(√3-1)](x - 2) and y - 3 = [(√3-1)/(√3+1)](x - 2); side length = √6</p>
Correct Answer: Equations: y - 3 = (m)(x - 2) with m = (√3+1)/(√3-1) or (√3-1)/(√3+1); side length = √6