A tangent is perpendicular to the radius

Builds on Tangent line to a circle Circle Vectors are perpendicular if and only if their dot product is zero

Statement

Let $C$ be a circle with the centre $O$ and the radius $r > 0$ in the Euclidean plane $\mathbb{R}^2$ . Let $T$ be a point of $C$ , and let $\ell$ be a line through $T$ . Then $\ell$ is tangent to $C$ at $T$ if and only if $\ell$ is perpendicular to the radius $OT$ . That is, $(X - T) \cdot (T - O) = 0$ for each point $X$ of $\ell$ .

Remarks

A proof: for $X = T + t v$ on $\ell$ , we have $\|X - O\|^2 = r^2 + 2t\, v \cdot (T - O) + t^2 \|v\|^2$ . So $t = 0$ is the only point of $\ell$ on $C$ if and only if $v \cdot (T - O) = 0$ . Thus each point of $C$ has exactly one tangent line. The same holds for a sphere and its tangent plane in $\mathbb{R}^3$ .

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Angle between two vectors concept
Circle concept
Length of a vector concept
Tangent line to a circle concept
Geometric formula for the dot product statement
Vectors are perpendicular if and only if their dot product is zero statement
A tangent is perpendicular to the radius statement

Tag	`031JV5S54D`
URI	`m2://031JV5S54D`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 031JV5S54D`.
Uses	Tangent line to a circle, Circle, Vectors are perpendicular if and only if their dot product is zero

A tangent is perpendicular to the radius

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here