A tangent is a line that just skims the surface of a circle. It hits the circle at one point only.

There are two main theorems that deal with tangents. The first one is as follows:

A tangent line of a circle will always be perpendicular to the radius of that circle. It will always form a right angle (90°) with the radius.


Questions that deal with this theorem usually go hand in hand with the Pythagorean theorem. That’s because you can only use this theorem if you have a right triangle. The Pythagorean theorem is: \({a^2} + {b^2} = {c^2}\) where “c” is always the hypotenuse.

Example 1:

Is \({CB}\) a tangent?

If \({CB}\) is a tangent, then that should be a right triangle which means the Pythagorean theorem will work.

Example 2:

\({CB}\) is a tangent. Find x.

\(\begin{array}{l}
{a^2} + {b^2} = {c^2}\\
{x^2} + {8^2} = {10^2}\\
{x^2} + 64 = 100\\
{x^2} = 36\\
x = 6
\end{array}\)

There is another theorem that deals with tangents as well.

If two tangents to the same circle share a point outside of the circle, then the two tangents are congruent.

If \({AP}\) and \({BP}\) are tangents, then \(\overline {AP}  \cong \overline {BP} \)

Example 1:

\(\overline {AB} \) and \({BC}\) are tangents. Find x.

Since both lines are tangents and share the point B, then they are equal.

x = 15

Example 2:

\(\overline {AB} \) and \({BC}\) are tangents. Find x.

These lines are equal as well.

\(\begin{array}{l}
\overline {AB}  \cong \overline {BC} \\
x - 24 = 50\\
x = 74
\end{array}\)

Let’s try a tougher one.

Example 3:

All lines are tangents. Find the perimeter of the polygon.

To find perimeter, add up all the   numbers. 8 + 8 + 3.9 + 3.9 + 8 + 8 + 3.9 + 3.9 = 47.6

Let’s try one last example.

Example 4:

All lines are tangents. Find the perimeter of the polygon.

This one is a little bit tougher. We have to figure it out piece by piece.

Now, we just have a few more pieces to   figure out.

Add all the sides. 10.3 + 10.3 + 6.1 + 6.1 + 13 + 13 + 8.8 + 8.8 = 76.4