How to tell if a line is tangent to a circle

A tangent line meets a circle at one point. We’ll draw the radius to that point to see how a right triangle can tell us whether the line is tangent and help us find a missing length.

A tangent is perpendicular to the radius at the point of contact

Draw a radius from the center to the point where the line touches the circle. The radius is perpendicular to the tangent there. The converse also works: if a line through a point on a circle is perpendicular to the radius at that point, the line is tangent.

To see why the converse works, pick any other point on that perpendicular line. Its distance to the center is the hypotenuse of a right triangle, which is longer than the radius. So that other point lies outside the circle.

Example checking whether AB is tangent

Circle $C$ has radius $5$ . Point $B$ is on the circle, $A B = 6$ , and $D$ lies between $C$ and $A$ on the same line, with $D A = 3$ . Since $C D = 5$ , the distance $C A$ is $8$ .

If AB were tangent at B, the radius CB would meet AB at a right angle.

For a right angle at B, the squares of CB and AB must add to the square of CA.

\begin{array}{l} C B^{2} + A B^{2} = 5^{2} + 6^{2} = 61 \\ C A^{2} = 8^{2} = 64 \end{array}

The squares don’t match, so the angle at B isn’t a right angle and AB isn’t tangent.

Example finding the radius from a tangent

Now $A B$ is tangent at $A$ , $A B = 15.2$ , and $D B = 7.6$ . Let the radius be $x$ . Because $C$ , $D$ , and $B$ are in that order, $C B = x + 7.6$ .

The radius AC and tangent AB meet at a right angle at A. Along the lower side, CB equals CD plus DB.

The right angle is at A, so CB is the hypotenuse. Use the Pythagorean theorem and expand the right side.

\begin{array}{l} x^{2} + {15.2}^{2} = {(x + 7.6)}^{2} \\ x^{2} + 231.04 = x^{2} + 15.2 x + 57.76 \\ 173.28 = 15.2 x \\ x = 11.4 \end{array}

The circle’s radius is 11.4 units.

Two tangents from the same point have equal lengths

If two tangent segments $A B$ and $A C$ touch a circle with center $O$ , the radii $O B$ and $O C$ are perpendicular to those segments. The two right triangles share hypotenuse $A O$ and have equal radii. Hypotenuse-leg congruence gives $A B = A C$ .

For example, if $A B = 3 x - 6$ and $A C = 15$ , set $3 x - 6 = 15$ and solve to get $x = 7$ . The same congruence also shows that $A O$ bisects the angle between the tangents.

To see the congruence rule used in the two tangents theorem, read how to prove triangles congruent using hypotenuse and leg.

Learn mathKrista KingJune 15, 2021math, learn online, online course, online math, geometry, circles, geometry of circles, tangent lines of circles, circle tangent lines, tangent lines, circle tangent line problems

How to tell if a line is tangent to a circle

A tangent is perpendicular to the radius at the point of contact

Example checking whether AB is tangent

Example finding the radius from a tangent

Two tangents from the same point have equal lengths

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