View source for Template:Right angle altitude.svg

[[File:right_angle_altitude.svg|thumb|right|The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using [[Pythagoras' theorem]] on the 3 triangles of sides {{nowrap|(''p''&thinsp;+&thinsp;''q'', ''r'', ''s''&#8202;)}}, {{nowrap|(''r'', ''p'', ''h''&#8202;)}} and {{nowrap|(''s'', ''h'', ''q''&#8202;)}},<br style="margin-bottom:1ex;" />
<math>\begin{align}
            (p + q)^2 \;\; &= \quad r^2 \;\;\, + \quad s^2 \\
 p^2 \!\!+\! 2pq \!+\! q^2 &= \overbrace{p^2 \!\!+\! h^2} + \overbrace{h^2 \!\!+\! q^2} \\
           2pq \quad\;\;\; &= 2h^2 \; \therefore h \!=\! \sqrt{pq} \\
\end{align}</math>]]