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2D Coordinate Geometry

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

Distance between Two Points

Given the line segment \( \overline{PQ} \) with \( P=(12,11) \) and \( Q=(8,5) \), the length of the line segment is \(a \).

What is \(a^2\)?

The points \(A,B,\) and \(C\) lie on a straight line such that the distance from \(A\) to \(B\) is 4 inches, and the distance from \(B\) to \(C\) is 5 inches. An ant crawled along the straight line from \(A\) to \(C\), and then back to \(B\). What is the total distance (in inches) that the ant travelled?

If \(A=(1,a)\), \(B=(4,2)\), and the distance between \(A\) and \(B\) is 5, what is the positive value of \(a?\)

What is the distance between \( (4, 8 ) \) and the origin?

Let \(a\) be a point on a number line such that its distance from the point \(3\) on the same number line is \(9.\) What are the possible values of \(a ?\)

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