Tangent Line Theorems
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; .
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?
Determine if the triangle below is a right triangle.
Use the Pythagorean Theorem. is the longest side, so it will be .
is not a right triangle. From this, we also find that is not tangent to .
If and are the centers and is tangent to both circles, find .
and and by AA Similarity.
To find , use the Pythagorean Theorem.
To find , use similar triangles.
is tangent to at point . Find . Reduce any radicals.
is tangent, so and a right triangle. Use the Pythagorean Theorem to find .
Using the answer from Example A above, find in . Round your answer to the nearest hundredth.
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.
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.
|4.||4. Definition of perpendicular lines|
|5.||5. Connecting two existing points|
|6. and are right triangles||6.|
- 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 .
To see the Review answers, open this PDF file and look for section 9.2.