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Distance Formula

The distance formula is derived by using the Pythagorean theorem.
Notice in the figure below that the distance between the points (x, y) and (a, b) is the hypotenuse of a right triangle. The difference y – b is the measure of the height of the triangle, and the difference x – a is the length of base of the triangle.
Applying the Pythagorean theorem yields
 
 
Taking the square root of both sides this equation yields
 

 
Example
In the figure, the circle is centered at the origin and passes through point P. Which one of the following points does it also pass through?
A.   (3, 3)
B.   
C.   (2, 6)
D.   
E.   (–3, 4)
 

Since the circle is centered at the origin and passes through the point (0, –3), the radius of the circle is 3.

 

Now, if any other point is on the circle, the distance from that point to the center of the circle (the radius) must also be 3. Look at choice (B).

 

Using the distance formula to calculate the distance between  and (0, 0) (the origin) yields

 

 

Hence,  is on the circle, and the answer is (B).
 





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