Metric Relationships in a Right Triangle

Properties

Consider the following right triangle:

• The Pythagorean theorem:

\(({\textrm{m}{\overline {\textrm{BC}}}})^{2}\) = \(({\textrm{m}{\overline {\textrm{AB}}}})^{2}\) + \(({\textrm{m}{\overline {\textrm{AC}}}})^{2}\)
\(\space\)

• The mean proportional:

\(({\textrm{m}{\overline {\textrm{AC}}}})^{2}\) = m\(\overline {\textrm{BC}}\space \times \)  m\(\overline {\textrm{HC}}\)
\(({\textrm{m}{\overline {\textrm{AB}}}})^{2}\) = m\(\overline {\textrm{BC}}\space \times \) m\(\overline {\textrm{HB}}\)
\(({\textrm{m}{\overline {\textrm{AH}}}})^{2}\) = m\(\overline {\textrm{HB}}\space \times\) m\(\overline {\textrm{HC}}\)
\(\space\)

• The product of the measures of the right angle’s sides:

m\(\overline {\textrm{AB}}\space \times\) m\(\overline {\textrm{AC}}\) = m\(\overline {\textrm{BC}}\space \times\) m\(\overline {\textrm{AH}}\)