9.2. Tangent Lines http://www.ck12.org
- Draw a tangent line,
←→
BC, whereBis the point of tangency. To draw a tangent line, take your ruler and line it
up with pointB. Make sure thatBis the only point on the circle that the line passes through.- Using your protractor, findm^6 ABC.
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.
To prove this theorem, the easiest way to do so is indirectly (proof by contradiction). Also, notice that this theorem
uses the words “if and only if,” making it a biconditional statement. Therefore, the converse of this theorem is also
true. Now let’s look at two tangent segments, drawn from the same external point. If we were to measure these two
segments, we would find that they are equal.
Two Tangents Theorem:If two tangent segments are drawn from the same external point, then the segments are
equal.
Example A
In
⊙
A,CBis tangent at pointB. FindAC. Reduce any radicals.
Solution:BecauseCBis tangent,AB⊥CB, making 4 ABCa right triangle. We can use the Pythagorean Theorem to
findAC.
52 + 82 =AC^2
25 + 64 =AC^2
89 =AC^2
AC=
√
89
Example B
FindDC, in
⊙
A. Round your answer to the nearest hundredth.Solution:
DC=AC−AD
DC=
√
89 − 5 ≈ 4. 43
Example C
Find the perimeter of 4 ABC.
Solution:AE=AD,EB=BF, andCF=CD. Therefore, the perimeter of 4 ABC= 6 + 6 + 4 + 4 + 7 + 7 =34.
We say that
⊙
Gisinscribedin 4 ABC. A circle is inscribed in a polygon, if every side of the polygon is tangent to
the circle.
Example D
Find the value ofx.