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Pythagorean Theorem vs. Square's Diagonal

Date: 03/19/2003 at 05:29:48
From: Meighan Doyle
Subject: Pythagorean theorem vs. square's diagonal

We learn the Pythagorean theorem, which states that the sum of the 
squares of each (non-hypotenuse)side of a right triangle equals the 
square of the hypotenuse. Using this, for a triangle with sides of 
length 1, we get a hypotenuse of the square root of 2. 

But imagine if you wanted to get from one corner of this triangle to 
the other via the hypotenuse, but could only make moves perpendicular 
to the other two sides. Let's say you move 1/2 right, 1/2 up, 1/2 
right, and 1/2 up. Your total trip length is 2. Go smaller. 1/4 right, 
1/4 up, etc. Your trip still adds to two. 

What if we did an infinite number of steps like this? Would the trip 
still equal 2? When does it magically turn into the SQUARE ROOT of 2?

Date: 03/19/2003 at 08:42:22
From: Doctor Peterson
Subject: Re: Pythagorean theorem vs. square's diagonal

Hi, Meighan.

Imagine you are driving down a long straight road; say it's a mile 

Now imagine you're a terrible driver, and rather than go straight down 
the middle of your lane, you zigzag constantly:

      /\        /\        /\
    o    \    /    \    /    \    o
           \/        \/        \/

Do you see that you would be going much farther than if you went 

This is what you are doing in trying to measure the diagonal by taking 
horizontal and vertical steps: you are not staying ON the diagonal, 
but zigzagging NEAR it, and increasing the distance traveled by a 
factor of sqrt(2).

This kind of thing does come up in calculus, where it is a common 
pitfall when people learn how to find the length of a curve.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Triangles and Other Polygons

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