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# compass and straightedge construction of perpendicular

Let $P$ be a point and $\ell$ be a line in the Euclidean plane. One can construct a line $m$ perpendicular to $\ell$ and passing through $P$. The construction given here yields $m$ in any circumstance: Whether $P\in\ell$ or $P\notin\ell$ does not matter. On the other hand, the construction looks quite different in these two cases. Thus, the sequence of pictures on the left (in which $\ell$ is in red) is for the case that $P\notin\ell$, and the sequence of pictures on the right (in which $\ell$ is in green) is for the case that $P\in\ell$.

1. With one point of the compass on $P$, draw an arc that intersects $\ell$ at two points. Label these as $Q$ and $R$.

2. Construct the perpendicular bisector of $\overline{QR}$. This line is $m$.

This construction is justified because $Q$ and $R$ are constructed so that $P$ is equidistant from them and thus lies on the perpendicular bisector of $\overline{QR}$.

In the case that $P\notin\ell$, this construction is referred to as *dropping the perpendicular* from $P$ to $\ell$. In the case that $P\in\ell$, this construction is referred to as *erecting the perpendicular* to $\ell$ at $P$.

If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.

## Mathematics Subject Classification

51M15*no label found*51-00

*no label found*

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