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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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