Up

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:
9257 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:
7639 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:
6506 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:

 
 
 

Facebook PageYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems