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# Pythagorean triples

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The numbers $20$, $99$, and $101$ form a Pythagorean triple. A right triangle has heights $x$, $\dfrac{20}{101}$, and $\dfrac{99}{101},$ where $x$ is the shortest height. What is $x?$

I know that the first sentence about the Pythagorean triple won't help. To solve for x, I take the longest side, square it, and subtract the square of the middle side. I get $\sqrt{\frac{99}{101}^2-\frac{20}{101}^2}$, which simplifies to $\sqrt{\frac{9401}{10201}}$. But my answer is wrong. What is the correct way to do this?

Feb 3, 2021

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The  hypotenuse    =    sqrt  [ (99^2  + 30^2  ]   /101  =   101/ 101 =   1

The  other altitude will  be  drawn to  this hypotenuse and  its length =  product  of the  legs/ hypotenuse  length  =

So    (20 / 101  *  99 /101)  / 1  = 1980 / 10201    ≈     .194  = x

Feb 3, 2021