Algebra 1 Distance Between Two Points
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Far Out
Distance Between Two Points

 
 What else could we do with two points?
 
 We could find the distance between them!
 
 Let's find the distance between 
 our old friends from the last page ...
 

 
 So how do we do it?
 
 We use something called the Pythagorean Theorem
 (What a name, eh?)
 
 Anyway, it works when you have a triangle 
 with a 90 degree angle (called a right triangle).
 
 The deal is, in a right triangle, 
 if you take the lengths of the two shorter sides
 multiply them times themselves and add that together
 you get the same answer as you do
 when you take the longest side of the triangle
 and multiply it times itself.
 
 In math talk ...
 

 
 Oh yeah?,
 Well I didn't see any triangles on the graph.
 

 Well then you're not looking hard enough.

 It's there ... really.
 One of the lines of the triangle,
 is the line between the two points.
 

 
 and a line that runs straight down 
 from the point on the right ...
 

 
 and a line that runs straight across 
 from the point on the left.
 

 
 If you put them together, you get ...
 

 
 Since one of these last two lines runs exactly up and down
 and the other one runs exactly left and right
 where they meet, they make a 90 degree angle.
 

 
 We also know that the point where the two lines meet
 has the X value of the point on the right
 and the Y value of the point on the left.
 Study the picture until you understand why.
 
 Now we can find the lengths of the two shorter sides.
 The one that runs from left to right 
 goes from the point (1,2) to the point (5,2)
 For both of these points, Y = 2.
 That means the distance is the difference 
 between the X coordinates ...
 

 
 We can use this trick to find the distance
 between the two points on the right side.
 
 The X values of these two points are both 5,
 only the Y values are different.
 The distance between these points is that difference.
 

 
 Now we know the lengths we need
 to use that Pythagorean Theorem thing.
 
 It will let us figure out the length 
 of the long side of the triangle.
 Which just happens to be the distance
 between the two points!
 

 

 
 This time, we don't have to worry about that +/- thing
 that you usually get with a root.
 That's because we're talking about a distance here.
 And this distance is not going to be negative number.
 
 So the distance between the points (1,2) and (5,3)
 is about 4.124.
 

   copyright 2005 Bruce Kirkpatrick

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