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Tangents

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

Circles Tangents 1

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.

Circles Tangents 2
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:

Circles Tangents 3 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:


Circles Tangents 4 \({CB}\) 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.


Circles Tangents 5 If \({AP}\) and \({BP}\) are tangents, then \(\overline {AP}  \cong \overline {BP} \)

 

Example 1:

Circles Tangents 6 \(\overline {AB} \) and \({BC}\) are tangents. Find x.

 

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

x = 15

Example 2:


Circles Tangents 7 \(\overline {AB} \) 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:

Circles Tangents 8

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.

Circles Tangents 9

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:

Circles Tangents 10

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.

Circles Tangents 11 Now, we just have a few more pieces to   figure out.

Circles Tangents 12 Add all the sides.

10.3 + 10.3 + 6.1 + 6.1 + 13 + 13 + 8.8 + 8.8 = 76.4


Downloads:
2618 x

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:

Circles-Tangents-Easy

Watch bellow how to solve this example:

 

Downloads:
2062 x

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:

Circles-Tangents-Medium

Watch bellow how to solve this example:

 

Downloads:
1840 x

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:

Circles-Tangents-Hard

Watch bellow how to solve this example:

 
 
 

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