Math Core

A wheel rolling on a flat road touches the road at exactly one point, and the spoke pointing straight down to that point stands perfectly upright. That picture captures the most important fact about tangent lines: a tangent is always perpendicular to the radius at the point where it touches. Combined with the Pythagorean theorem, this lets you find distances in and around circles.

Tangents are perpendicular to radii

Recall that a tangent is a line in the plane of a circle that meets the circle at exactly one point, the point of tangency. A tangent segment is a segment of a tangent line with one endpoint at the point of tangency.

Tangent–radius theorem

A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency.

If line $\ell$ is tangent to $\odot O$ at $T$ , then $\overline{OT} \perp \ell$ . And if a line through a point $T$ on the circle is perpendicular to $\overline{OT}$ , it is tangent.

Here is why. Suppose line $\ell$ touches $\odot O$ only at $T$ . Every other point of $\ell$ is outside the circle, so it is farther from $O$ than $T$ is. That makes $T$ the closest point on $\ell$ to $O$ , and the shortest segment from a point to a line is the perpendicular one. So $\overline{OT} \perp \ell$ .

Right triangles from tangents

When you draw a tangent segment $\overline{PT}$ from an outside point $P$ , then draw the radius $\overline{OT}$ and the segment $\overline{OP}$ , you get a right triangle with the right angle at $T$ . The segment $\overline{OP}$ is the hypotenuse.

PT and PU are tangent to circle O. Radius OT = 3 and tangent segment PT = 4, so OP = 5.

Worked example: Finding a distance

A circle has radius $5$ . A tangent segment from point $P$ to the circle has length $12$ . How far is $P$ from the center?

The radius, the tangent segment and $\overline{OP}$ form a right triangle with hypotenuse $\overline{OP}$ :

$OP = \sqrt{5^2 + 12^2} = \sqrt{169} = 13.$

You can also run this backward to test whether a line is tangent: check whether the three lengths satisfy the Pythagorean theorem.

Worked example: Is it tangent?

$\odot O$ has radius $6$ . Point $T$ is on the circle, $PT = 8$ and $OP = 10$ . Is $\overline{PT}$ tangent to the circle?

Check the converse of the Pythagorean theorem: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ . So $\triangle OTP$ is a right triangle with the right angle at $T$ . Since $\overline{PT} \perp \overline{OT}$ , $\overline{PT}$ is tangent.

Common mistake

The hypotenuse is the segment from the outside point to the center, not the tangent segment. A common mistake is to write $5^2 + 13^2$ when the $13$ is really the hypotenuse. Always put the right angle at the point of tangency, then identify the side across from it.

Two tangents from the same point

From a point outside a circle, you can draw exactly two tangents. In the figure above, $\overline{PT}$ and $\overline{PU}$ both start at $P$ . Triangles $OTP$ and $OUP$ are both right triangles with the same hypotenuse $\overline{OP}$ and congruent legs $\overline{OT} \cong \overline{OU}$ (radii). By HL they are congruent, which gives this theorem.

Tangent segments theorem

If two segments from the same exterior point are tangent to a circle, they are congruent: $PT = PU$ .

Worked example: Tangent segments with algebra

$\overline{PA}$ and $\overline{PB}$ are tangent to a circle at $A$ and $B$ . If $PA = 2x + 3$ and $PB = 5x - 9$ , find $PA$ .

The tangent segments are equal: $2x + 3 = 5x - 9$ , so $12 = 3x$ and $x = 4$ . Then $PA = 2(4) + 3 = 11$ .

This theorem also explains triangles with a circle inscribed inside them. Each vertex sends two tangent segments to the circle, and those two are equal. So if you know one tangent length from each vertex, you know all six pieces of the perimeter.

The angle between a tangent and a chord

An angle can also be formed by a tangent and a chord that meet at the point of tangency. It acts like an inscribed angle whose vertex slid onto the circle.

Tangent–chord angle

If a tangent and a chord meet at a point on the circle, the angle they form is half the intercepted arc.

For example, if a tangent at $A$ and chord $\overline{AB}$ form a $70^\circ$ angle, the arc inside that angle measures $140^\circ$ .

Tip

When two tangents from $P$ touch a circle at $T$ and $U$ , quadrilateral $OTPU$ has two right angles. Its other two angles add to $360^\circ - 180^\circ = 180^\circ$ , so $m\angle TPU + m\angle TOU = 180^\circ$ .