
# 1.2: Distance Between Two Points; Circles

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Given two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$, recall that their horizontal distance from one another is $$\Delta x=x_2-x_1$$ and their vertical distance from one another is $$\Delta y=y_2-y_1$$. (Actually, the word "distance'' normally denotes "positive distance''. $$\Delta x$$ and $$\Delta y$$ are signed distances, but this is clear from context.) The actual (positive) distance from one point to the other is the length of the hypotenuse of a right triangle with legs $$|\Delta x|$$ and $$|\Delta y|$$, as shown in Figure $$\PageIndex{1}$$. The Pythagorean theorem then says that the distance between the two points is the square root of the sum of the squares of the horizontal and vertical sides:

$$\hbox{distance} =\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+ (y_2-y_1)^2}.$$

For example, the distance between points $$A(2,1)$$ and $$B(3,3)$$ is

$$\sqrt{(3-2)^2+(3-1)^2}=\sqrt{5}.$$