<p>A parallelogram has vertices at <em>A</em>(0, 0), <em>C</em>(4, 4), and the midpoint of diagonal <em>AC</em> is <em>E</em>(2, 2). The locus of the extremities of the other diagonal is:</p>
Step-by-Step Solution
Key Concept: In a parallelogram, the diagonals bisect each other at the same point. Since E(2,2) is the midpoint of AC, it must also be the midpoint of the other diagonal BD, meaning if B(x₁,y₁) and D(x₂,y₂) are endpoints, then (x₁+x₂)/2 = 2 and (y₁+y₂)/2 = 2.
<p><strong>Step 1:</strong> In a parallelogram, diagonals bisect each other. Here AC is one diagonal with A(0,0) and C(4,4), and its midpoint is E(2,2).</p><p><strong>Step 2:</strong> Let B(x,y) be one vertex of the other diagonal BD. Since E(2,2) must be the midpoint of BD, if D has coordinates (x',y'), then: (x+x')/2 = 2 and (y+y')/2 = 2, giving x' = 4-x and y' = 4-y.</p><p><strong>Step 3:</strong> As B(x,y) varies over all points in the plane (except positions that would make ABCD degenerate), the point D is determined by D(4-x, 4-y). The locus of B is the entire plane excluding only the line AC itself (since B and D cannot lie on AC).</p><p><strong>Step 4:</strong> However, if the question asks for the locus as a geometric constraint: B and D must satisfy that their midpoint is (2,2). This means for any choice of B, D is automatically determined. The answer represents the constraint surface: <strong>x + x' = 4 and y + y' = 4</strong>, which describes that one extremity can be any point and the other is its reflection through E(2,2).</p><p>∴ Answer: The locus is all points such that their reflection through (2,2) forms the other diagonal endpoint</p>
Correct Answer: 8