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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.

In the circle $O$ , $\stackrel{\leftrightarrow }{PT}$ is a tangent and $\overline{OP}$ is the radius.

If $\stackrel{\leftrightarrow }{PT}$ is a tangent, then $\overline{OP}$ is perpendicular to $\stackrel{\leftrightarrow }{PT}$ .

For example, suppose $\overline{OP}$ $=3$ units and $\overline{PT}$ $=4$ units. Find the length of $\overline{OT}$ .

Because the radius is perpendicular to the tangent at the point of tangency, $\overline{OP}\perp \stackrel{\leftrightarrow }{PT}$ .

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

Now use the Pythagorean Theorem to find $\overline{OT}$ .

$\begin{array}{l}{\left(OP\right)}^2+{\left(PT\right)}^2={\left(OT\right)}^2\\ 3^2+4^2={\left(OT\right)}^2\\ 9+16={\left(OT\right)}^2\\ 25={\left(OT\right)}^2\\ \pm 5=OT\end{array}$

Since the length cannot be negative, the length of $\overline{OT}$ is $5$ units.