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In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.

\((4, 5)\) is the midpoint between \((2, 1)\) and \((x, y).\) What is the value of \(xy?\)

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A bug sitting at the point \((1, 2)\) moves 5 units vertically (parallel to the \(y\)-axis) and 1 units horizontally (parallel to the \(x\)-axis). Which ordered pair gives potential coordinates for the bug's new location?

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The points \(A, B, C,\) and \(D\) form a rectangle that is twice as tall as it is wide. Given the coordinates of two of the points \(A = (2, -5),\) and \(B = (2, 1),\) where might another point of the rectangle (\(C\) or \(D\)) be located?

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\(Q = (x, y),\) is a point on line segment \(BC\) such that, comparing the lengths, \(BQ:QC = 1:3.\) What is the value of \(x + y\) if \(B\) is the point (2, 2), and \(C,\) the point (6, 14).

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Circle \(C\) has a center at \((2, 1)\) with a radius of 5. Which point could lie on the circle?

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