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'hypotenuse'
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| Title of object: |
hypotenuse |
| Canonical Name: |
Hypotenuse |
| Type: |
Definition |
| Created on: |
2001-12-12 04:09:21 |
| Modified on: |
2007-07-06 14:45:08 |
| Classification: |
msc:51-00 |
| Synonyms: |
hypotenuse=hypothenuse |
Revision comment (for changes between this and next version):
| Changes for correction #12142 ('isosceles triangles'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $ABC$ a right triangle in a Euclidean geometry with right angle at $C$. Then $AB$ is called the \emph{hypotenuse} of $ABC$.
\begin{center}
\includegraphics{hyp}
\end{center}
The middle point $P$ of the hypotenuse coincides with the circumcenter of the triangle, so it is equidistant from the three vertices. When the triangle is inscribed on his circumcircle, the hypotenuse becomes a diameter. So the distance from $P$ to the vertices is precisely the circumradius.
The hypotenuse's length can be calculated by means of the Pythagorean theorem:
$$c=\sqrt{a^2+b^2}$$
\textbf{Remark}. Sometimes, the longest side of a triangle is also called an hypotenuse, but this naming is seldom seen. |
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