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.

Is \({CB}\) a tangent? |

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

\(\begin{array}{l} {a^2} + {b^2} = {c^2}\\ {3^2} + {12^2} = {15^2}\\ 9 + 144 = 225\\ 153 \ne 225 \end{array}\) |
It doesn’t work, so \({CB}\) is not a tangent! |

**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. |

We have to determine which lines are equal. They have to be tangents that hit the same point.

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 |

Determine if line AB is tangent to the circle.

This free worksheet contains 10 assignments each with 24 questions with answers.

**Example of one question:**

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Find the perimeter of each polygon. Assume that lines which appear to be tangent are tangent.

This free worksheet contains 10 assignments each with 24 questions with answers.

**Example of one question:**

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Solve for x. Assume that lines which appear to be tangent are tangent.

This free worksheet contains 10 assignments each with 24 questions with answers.

**Example of one question:**

**Watch bellow how to solve this example:**