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2D Coordinate Geometry

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

Coordinate Geometry Warmup


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

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?

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?

\(Q = (x, y),\) 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).

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


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