<p>Each side of a square is of length 4 units. The center of the square is at (3, 7) and one of the diagonals is parallel to the line y = x. If the vertices of the square be (x₁, y₁), (x₂, y₂), (x₃, y₃) and (x₄, y₄), then find the value of max(y₁, y₂, y₃, y₄) - min(x₁, x₂, x₃, x₄).</p>
Step-by-Step Solution
Key Concept: Since one diagonal is parallel to y = x (slope = 1), the other diagonal is perpendicular to it (slope = -1). The diagonals of a square bisect each other at the center and are equal in length. For a square with side 4, each diagonal has length 4√2, so vertices are at distance 2√2 from the center along these diagonal directions.
<p><strong>Step 1: Set up the coordinate system.</strong> The center of the square is at C = (3, 7). One diagonal is parallel to y = x, which has slope 1. The other diagonal is perpendicular, with slope -1.</p><p><strong>Step 2: Find the direction vectors of the diagonals.</strong> The diagonal parallel to y = x has direction vector proportional to (1, 1), normalized to (1/√2, 1/√2). The perpendicular diagonal has direction vector (1/√2, -1/√2).</p><p><strong>Step 3: Calculate distance from center to vertex.</strong> For a square with side length a = 4, the diagonal length is 4√2. Each vertex is at distance (4√2)/2 = 2√2 from the center.</p><p><strong>Step 4: Find the four vertices.</strong> The vertices lie along the two perpendicular diagonals at distance 2√2 from the center:<br/>• Along diagonal with slope 1: moving ±2√2 in direction (1/√2, 1/√2)<br/> - Vertex A: (3, 7) + 2√2(1/√2, 1/√2) = (3, 7) + (2, 2) = (5, 9)<br/> - Vertex C: (3, 7) - 2√2(1/√2, 1/√2) = (3, 7) - (2, 2) = (1, 5)<br/>• Along diagonal with slope -1: moving ±2√2 in direction (1/√2, -1/√2)<br/> - Vertex B: (3, 7) + 2√2(1/√2, -1/√2) = (3, 7) + (2, -2) = (5, 5)<br/> - Vertex D: (3, 7) - 2√2(1/√2, -1/√2) = (3, 7) - (2, -2) = (1, 9)</p><p><strong>Step 5: Identify the four vertices.</strong> The vertices are: (5, 9), (5, 5), (1, 5), (1, 9).</p><p><strong>Step 6: Find max(y-coordinates) and min(x-coordinates).</strong><br/>y-coordinates: 9, 5, 5, 9 → max(y₁, y₂, y₃, y₄) = 9<br/>x-coordinates: 5, 5, 1, 1 → min(x₁, x₂, x₃, x₄) = 1</p><p><strong>Step 7: Calculate the answer.</strong> max(y₁, y₂, y₃, y₄) - min(x₁, x₂, x₃, x₄) = 9 - 1 = 8</p><p><strong>∴ Answer: 8</strong></p>
Correct Answer: 8