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Pythagorean Reciprocals (Posted on 2020-02-05) Difficulty: 2 of 5
Let a and b be the lengths of the two legs of a right triangle and let c be the length of the hypotenuse. The Pythagorean Theorem states that a^2 + b^2 = c^2.

If the reciprocals of the legs are used instead to form 1/a^2 + 1/b^2 = 1/x^2, then what is special about x?

No Solution Yet Submitted by Brian Smith    
Rating: 2.0000 (1 votes)

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thoughts | Comment 1 of 5
x is the length of the line dropped perpendicularly from the hypotenuse to the vertex of the right angle. Why?

If we place the right angle at the origin, the hypotenuse has x-intercept at a and y-intercept at b, then the equation of the hypotenuse as a line is y= (-b/a) x + b

or 

x/a + y/b =1, which is of the form: Ax + By + C =0

I learned from the internet: 

The length d of the perpendicular from a such a line to an arbitrary point x1, y1 (A formula which can be shown with dot products) is

d = | (A x1 + B y1 + C) / Sqrt (A^2 + B^2) |

Here, this distance reduces (with x1 = y1 = 0 and C= -1)

to 

d = 1/sqrt(1/a^2 + 1/b^2) which yields d=x


Edited on February 5, 2020, 11:17 am
  Posted by Steven Lord on 2020-02-05 08:56:31

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