What if a line were drawn outside a circle that appeared to touch the circle at only one point? How could you determine if that line were actually a tangent? After completing this Concept, you'll be able to apply theorems to solve tangent problems like this one.
Watch This
First watch this video.
James Sousa: Tangent Lines to a Circle
Now watch this video.
James Sousa: Tangent Lines to a Circle Example Problems
Guidance
There are two important theorems about tangent lines.
1) Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
is tangent at point if and only if .
This theorem uses the words “if and only if,” making it a biconditional statement, which means the converse of this theorem is also true.
2) Two Tangents Theorem: If two tangent segments are drawn to one circle from the same external point, then they are congruent.
and have as an endpoint and are tangent; .
Example A
is tangent to at point . Find . Reduce any radicals.
is tangent, so and a right triangle. Use the Pythagorean Theorem to find .
Example B
Using the answer from Example A above, find in . Round your answer to the nearest hundredth.
Example C
Find the perimeter of .
and . Therefore, the perimeter of .
is inscribed in . A circle is inscribed in a polygon if every side of the polygon is tangent to the circle.
Guided Practice
1. Find between and . Reduce the radical.
2. Determine if the triangle below is a right triangle.
3. If and are the centers and is tangent to both circles, find .
4. Find the value of .
Answers:
1. and . Draw in , so is a rectangle. Use the Pythagorean Theorem to find .
2. Again, use the Pythagorean Theorem. is the longest side, so it will be .
Does
is not a right triangle. From this, we also find that is not tangent to .
3. and and by AA Similarity.
To find , use the Pythagorean Theorem.
To find , use similar triangles.
4. by the Two Tangents Theorem. Set and solve for .
Practice
Determine whether the given segment is tangent to .
Find the value of the indicated length(s) in . and are points of tangency. Simplify all radicals.
and are points of tangency for and .
 Is ? Why?
 Find .
 Find .
 Find .
 Find and .
is inscribed in .
 Find the perimeter of .
 What type of quadrilateral is ? How do you know?
 Draw a circle inscribed in a square. If the radius of the circle is 5, what is the perimeter of the square?
 Can a circle be inscribed in a rectangle? If so, draw it. If not, explain.
 Draw a triangle with two sides tangent to a circle, but the third side is not.
 Can a circle be inscribed in an obtuse triangle? If so, draw it. If not, explain.
 Fill in the blanks in the proof of the Two Tangents Theorem.
Given : and with points of tangency at and . and are radii.
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3. and  3. 
4.  4. Definition of perpendicular lines 
5.  5. Connecting two existing points 
6. and are right triangles  6. 
7.  7. 
8.  8. 
9.  9. 

Fill in the blanks, using the proof from #21.
 is a _____________ (type of quadrilateral).
 The line that connects the ___________ and the external point __________ .
 Points and are points of tangency for the three tangent circles. Explain why .