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Tangent to a Circle

A tangent to a circle is a straight line which touches the circle at only one point. This point is called the point of tangency.

The tangent to a circle is perpendicular to the radius at the point of tangency.

Math diagram

In the circle $O$ , $\overset{\leftrightarrow}{P T}$ is a tangent and $\bar{O P}$ is the radius.

If $\overset{\leftrightarrow}{P T}$ is a tangent, then $\bar{O P}$ is perpendicular to $\overset{\leftrightarrow}{P T}$ .

For example, suppose $\bar{O P}$ $= 3$ units and $\bar{P T}$ $= 4$ units. Find the length of $\bar{O T}$ .

Because the radius is perpendicular to the tangent at the point of tangency, $\bar{O P} ⊥ \overset{\leftrightarrow}{P T}$ .

This makes the angle $P$ a right angle in the triangle $O P T$ and triangle $O P T$ a right triangle.

Now use the Pythagorean Theorem to find $\bar{O T}$ .

$\begin{array}{l} {(O P)}^{2} + {(P T)}^{2} = {(O T)}^{2} \\ 3^{2} + 4^{2} = {(O T)}^{2} \\ 9 + 16 = {(O T)}^{2} \\ 25 = {(O T)}^{2} \\ \pm 5 = O T \end{array}$

Since the length cannot be negative, the length of $\bar{O T}$ is $5$ units.