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\begin{document}

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\section*{similarity}

\hypertarget{similarity}{}\section*{{Similarity}}\label{similarity}

Two shapes are \textbf{similar} if one can be obtained from the other by a series of rotations, reflections, translations, and scaling.

If two shapes are similar then the following hold:

\begin{itemize}%
\item Corresponding angles on the two shapes are equal.
\item There is a fixed scale factor such that corresponding lengths on the two shapes are related by multiplying by this scale factor.
\item The ratio of two lengths in one shape is the same as the ratio of the corresponding lengths in the other.

\end{itemize}
\hypertarget{similar_triangles}{}\subsection*{{Similar Triangles}}\label{similar_triangles}

When the two shapes are triangles then the following statements are equivalent:

\begin{itemize}%
\item The two triangles are similar
\item The angles in one triangle are the same as the angles in the other
\item The ratios of the lengths in one triangle are the same as in the other

\end{itemize}
\hypertarget{scale_factors}{}\subsection*{{Scale Factors}}\label{scale_factors}

When two shapes are similar, corresponding lengths are related by a fixed \textbf{scale factor}. Corresponding areas are also related by a scale factor, and if the shape is three dimensional so are volumes. These three scale factors are related as follows:

If the length scale factor is $k$, then the area scale factor is $k^2$, and the volume scale factor is $k^3$.

[[!redirects similar triangle]] [[!redirects similar triangles]] [[!redirects similar shape]] [[!redirects similar shapes]] [[!redirects similar]] [[!redirects scale factor]] [[!redirects length scale factor]] [[!redirects area scale factor]] [[!redirects volume scale factor]] [[!redirects scale factors]] [[!redirects length scale factors]] [[!redirects area scale factors]] [[!redirects volume scale factors]]



\end{document}