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How do we find the reduced radical form of the perimeter of a rectangle where only the length of the diagonal line between opposite vertices is known, only using the Pythagorean Theorem?

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We can't, because it isn't uniquely determined. For example, a rectangle with sides $7$ and $24$ has an opposite-vertices length of $\sqrt{49+576}=\sqrt{625}=25$, by the Pythagorean theorem, and it has a perimeter of $7+7+24+24=62$. But a square of side length $\frac{25}{\sqrt{2}}$ also has an opposite-vertices length of $\sqrt{\frac{625}{2}+\frac{625}{2}}=\sqrt{625}=25$, and it has a perimeter of $\frac{25}{\sqrt{2}}+\frac{25}{\sqrt{2}}+\frac{25}{\sqrt{2}}+\frac{25}{\sqrt{2}}=\frac{100}{\sqrt{2}}$.

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    That's correct, there are infinitely many solutions. Note that any rectangle with vertices at the origin and a point on the unit circle has an opposite-vertices length of 1, however some of these have a perimeter infinitely close to 2, while others have a perimeter of $2\sqrt{2}$.2011-10-09
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The perimeter of a rectangular window is 36 feet. The width of the window is 6 feet more than the length. What are the dimensions of the window?