Problem

Use coordinate evidence to prove that quadrilateral \(A(0,0)\), \(B(4,1)\), \(C(6,5)\), \(D(2,4)\) is a parallelogram.

Source coordinate plane: Quadrilateral ABCD has vertices A(0,0), B(4,1), C(6,5), and D(2,4); no slopes or parallelogram conclusion are shown.

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What this problem is really about

Pick one complete coordinate test for a parallelogram and carry it through; here, slopes make the opposite-side structure easy to track. Pair opposite sides rather than adjacent ones, compute each pair with consistent subtraction, and require both opposite pairs to have matching directions before invoking the definition.

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AB and CD both have slope 1/4, and BC and AD both have slope 2, so both pairs of opposite sides are parallel and ABCD is a parallelogram.

AB has slope 1/4 and BC has slope 2, so the sides have different slopes and that means ABCD is not a parallelogram.

AB has slope 1/4, BC has slope 2, CD has slope -1/4, and AD has slope 2, so only one pair of opposite sides is parallel and ABCD is not a parallelogram.

AB has slope 1/4 and BC has slope 2, so the sides are perpendicular, which proves ABCD is a parallelogram.

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Use coordinate evidence to prove that quadrilateral \(A(0,0)\), \(B(4,1)\), \(C(6,5)\), \(D(2,4)\) is a parallelogram.

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