6.18: Tangent Lines

Lines perpendicular to the radius drawn to the point of tangency.

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.

Figure \(\PageIndex{1}\)

\(\overleftrightarrow{BC}\) is tangent at point \(B\) if and only if \(\overleftrightarrow{BC}\perp \overline{AB}\).

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.

Figure \(\PageIndex{2}\)

\(\overline{BC}\) and \(\overline{DC}\) have \(C\) as an endpoint and are tangent; \(\overline{BC}\cong \overline{DC}\).

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?

Example \(\PageIndex{1}\)

Determine if the triangle below is a right triangle.

Figure \(\PageIndex{3}\)

Solution

Use the Pythagorean Theorem. \(4\sqrt{10}\) is the longest side, so it will be \(c\).

Does

\(\begin{aligned} 8^2+10^2&= (4\sqrt{10})^2? \\ 64+100&\neq 160\end{aligned}\)

\(\Delta ABC\) is not a right triangle. From this, we also find that \(\overline{CB}\) is not tangent to \(\bigodot A\).

Example \(\PageIndex{2}\)

If \(D\) and \(C\) are the centers and \(AE\) is tangent to both circles, find \(DC\).

Figure \(\PageIndex{4}\)

Solution

\(\overline{AE}\perp \overline{DE}\) and \(\overline{AE}\perp \overline{AC}\) and \(\Delta ABC \sim \Delta DBE\) by AA Similarity.

To find \(DB\), use the Pythagorean Theorem.

\(\begin{aligned} 10^2+24^2&=DB^2 \\ 100+576&=676 \\ DB&=\sqrt{676}=26 \end{aligned}\)

To find \(BC\), use similar triangles.

\(\dfrac{5}{10}=\dfrac{BC}{26}\rightarrow BC=13. \: DC=DB+BC=26+13=39\)

Example \(\PageIndex{3}\)

\(\overline{CB}\) is tangent to \(\bigodot A\) at point \(B\). Find \(AC\). Reduce any radicals.

Figure \(\PageIndex{5}\)

Solution

\(\overline{CB}\) is tangent, so \(\overline{AB}\perp \overline{CB}\) and \(\Delta ABC\) a right triangle. Use the Pythagorean Theorem to find \(AC\).

\(\begin{aligned} 5^2+8^2&=AC^2 \\ 25+64&=AC^2 \\ 89&=AC^2 \\ AC&=\sqrt{89}\end{aligned}\)

Example \(\PageIndex{4}\)

Using the answer from Example A above, find \(DC\) in \(\bigodot A\). Round your answer to the nearest hundredth.

Figure \(\PageIndex{6}\)

Solution

\(\begin{aligned} DC&=AC−AD \\ DC&=\sqrt{89}−5 \approx 4.43 \end{aligned}\)

Example \(\PageIndex{5}\)

Find the perimeter of \(\Delta ABC\).

Figure \(\PageIndex{7}\)

Solution

\(AE=AD\), \(EB=BF\), and \(CF=CD\). Therefore, the perimeter of

\(\Delta ABC=6+6+4+4+7+7=34\).

\(\bigodot G\) is inscribed in \(\Delta ABC\). A circle is inscribed in a polygon if every side of the polygon is tangent to the circle.

Review

Determine whether the given segment is tangent to \(\bigodot K\).

1. 

   Figure \(\PageIndex{8}\)

2. 

   Figure \(\PageIndex{9}\)

3. 

   Figure \(\PageIndex{10}\)

Find the value of the indicated length(s) in \(\bigodot C\). \(A\) and \(B\) are points of tangency. Simplify all radicals.

4. 

   Figure \(\PageIndex{11}\)

5. 

   Figure \(\PageIndex{12}\)

6. 

   Figure \(\PageIndex{13}\)

7. 

   Figure \(\PageIndex{14}\)

8. 

   Figure \(\PageIndex{15}\)

9. 

   Figure \(\PageIndex{16}\)

\(A\) and \(B\) are points of tangency for \(\bigodot C\) and \(\bigodot D\).

Figure \(\PageIndex{17}\)

10. Is \(\Delta AEC \sim \Delta BED\)? Why?
11. Find \(CE\).
12. Find \(BE\).
13. Find \(ED\).
14. Find \(BC\) and \(AD\).

\(\bigodot A\) is inscribed in \(BDFH\).

Figure \(\PageIndex{18}\)

15. Find the perimeter of \(BDFH\).
16. What type of quadrilateral is \(BDFH\)? How do you know?
17. Draw a circle inscribed in a square. If the radius of the circle is 5, what is the perimeter of the square?
18. Can a circle be inscribed in a rectangle? If so, draw it. If not, explain.
19. Draw a triangle with two sides tangent to a circle, but the third side is not.
20. Can a circle be inscribed in an obtuse triangle? If so, draw it. If not, explain.
21. Fill in the blanks in the proof of the Two Tangents Theorem.

Given: \(\overline{AB}\) and \(\overline{CB}\) with points of tangency at \(A\) and \(C\). \(\overline{AD}\) and \(\overline{DC}\) are radii.

Prove: \(\overline{AB}\cong \overline{CB}\)

Statement	Reason
1.	1.
2. \(\overline{AD}\cong \overline{DC}\)	2.
3. \(\overline{DA}\perp \overline{AB}\) and \(\overline{DC}\perp \overline{CB}\)	3.
4.	4. Definition of perpendicular lines
5.	5. Connecting two existing points
6. \(\Delta ADB\) and \(\Delta DCB\) are right triangles	6.
7.\(\overline{DB}\cong\overline{DB}\)	7.
8.\(\Delta ABD\cong \Delta CBD\)	8.
9.\(\overline{AB}\cong \overline{CB}\)	9.

22. Fill in the blanks, using the proof from #21.
    1. \(ABCD\) is a _____________ (type of quadrilateral).
    2. The line that connects the ___________ and the external point \(B\) __________ \(\angle ABC\).
23. Points \(A\), \(B\), and \(C\) are points of tangency for the three tangent circles. Explain why \(\overline{AT}\cong \overline{BT}\cong \overline{CT}\).

    

    Figure \(\PageIndex{19}\)

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.2.

Resources

Vocabulary

Term	Definition
circle	The set of all points that are the same distance away from a specific point, called the center.
diameter	A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
point of tangency	The point where the tangent line touches the circle.
radius	The distance from the center to the outer rim of a circle.
Tangent	The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.
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.
Two Tangent Theorem	The Two-Tangent Theorem states that if two tangent segments are drawn to one circle from the same external point, then they are congruent.

Additional Resources

Interactive Element

Video: Tangent Lines Principles - Basic

Activities: Tangent Lines Discussion Questions

Study Aids: Properties of a Circle Study Guide

Practice: Tangent Lines

Real World: Swing Rides