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The Almost Pythagorean Theorem (Posted on 2017-02-03) Difficulty: 3 of 5
A pretty good approximation of the Pythagorean Theorem to find the hypotenuse of a right triangle is to add the length of the longer leg to half the length of the shorter leg.

For what ratio of leg lengths is this approximation worst? How bad is it?

No Solution Yet Submitted by Jer    
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Some Thoughts Solution only. Comment 1 of 1

The worst approximation occurs when the long leg is twice the length of the short. At that point, the error is as sqrt(5)/2  is to 1.

When the long leg is almost as long as the hypotenuse, the shorter leg more or less falls out of the picture and the approximation becomes increasingly accurate.

To complete the picture, when the legs are equal, the error is as 3/(2sqrt(2)) is to 1.

  Posted by broll on 2017-02-04 00:10:45
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