Asked by Big Papa
A parallelogram has vertices A(0, 4), B(2, 2), C(4, 4), and D(2,6). Is this parallelogram a square? Explain why or why not.
All Answers 1
Answered by
GPT-5 mini
AI
Yes. Compute side vectors (or slopes and lengths):
- AB = B - A = (2, -2). BC = C - B = (2, 2).
- |AB| = sqrt(2^2 + (-2)^2) = sqrt(8) = 2√2; |BC| = sqrt(2^2 + 2^2) = 2√2. Similarly CD and DA have the same length, so all four sides are equal.
- AB · BC = (2)(2) + (-2)(2) = 4 - 4 = 0, so AB is perpendicular to BC (slope AB = -1, slope BC = 1).
A parallelogram with all sides equal and one right angle is a square. Therefore the given parallelogram is a square.
- AB = B - A = (2, -2). BC = C - B = (2, 2).
- |AB| = sqrt(2^2 + (-2)^2) = sqrt(8) = 2√2; |BC| = sqrt(2^2 + 2^2) = 2√2. Similarly CD and DA have the same length, so all four sides are equal.
- AB · BC = (2)(2) + (-2)(2) = 4 - 4 = 0, so AB is perpendicular to BC (slope AB = -1, slope BC = 1).
A parallelogram with all sides equal and one right angle is a square. Therefore the given parallelogram is a square.
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