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Tangent Lines

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Tangent Lines

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.

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CK-12 Foundation: Chapter9TangentLinesA

James Sousa: Tangent Lines to a Circle

James Sousa: Tangent Lines to a Circle Example Problems


The tangent line and the radius drawn to the point of tangency have a unique relationship. Let’s investigate it here.

Investigation: Tangent Line and Radius Property

Tools needed: compass, ruler, pencil, paper, protractor

  1. Using your compass, draw a circle. Locate the center and draw a radius. Label the radius \overline{AB} , with A as the center.
  2. Draw a tangent line, \overleftrightarrow{BC} , where B is the point of tangency. To draw a tangent line, take your ruler and line it up with point B . Make sure that B is the only point on the circle that the line passes through.
  3. Using your protractor, find m \angle 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 \bigodot A, \overline{CB} is tangent at point B . Find AC . Reduce any radicals.

Solution: Because \overline{CB} is tangent, \overline{AB} \bot \overline{CB} , making \triangle ABC a right triangle. We can use the Pythagorean Theorem to find AC .

5^2+8^2 &= AC^2\\25+64 &= AC^2\\89 &= AC^2\\AC &= \sqrt{89}

Example B

Find DC , in \bigodot A . Round your answer to the nearest hundredth.


DC &= AC - AD\\DC &= \sqrt{89}-5 \approx 4.43

Example C

Find the perimeter of \triangle ABC .

Solution: AE = AD, EB = BF , and CF = CD . Therefore, the perimeter of \triangle ABC=6+6+4+4+7+7=34 .

We say that \bigodot G is inscribed in \triangle ABC . A circle is inscribed in a polygon, if every side of the polygon is tangent to the circle.

Example D

Find the value of x .

Because \overline{AB} \bot \overline{AD} and \overline{DC} \bot \overline{CB}, \overline{AB} and \overline{CB} are tangent to the circle and also congruent. Set AB = CB and solve for x .

4x-9 &= 15\\4x &= 24\\x &= 6

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter9TangentLinesB

Guided Practice

1. Determine if the triangle below is a right triangle. Explain why or why not.

2. Find the distance between the centers of the two circles. Reduce all radicals.

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


1. To determine if the triangle is a right triangle, use the Pythagorean Theorem. 4 \sqrt{10} is the longest length, so we will set it equal to c in the formula.

8^2+10^2 & \ ? \ \left( 4 \sqrt{10} \right)^2\\64+100 &\neq 160

\triangle ABC is not a right triangle. And, from the converse of the Tangent to a Circle Theorem, \overline{CB} is not tangent to \bigodot A .

2. The distance between the two circles is AB . They are not tangent, however, \overline{AD} \bot \overline{DC} and \overline{DC} \bot \overline{CB} . Let’s add \overline{BE} , such that EDCB is a rectangle. Then, use the Pythagorean Theorem to find AB .

5^2+55^2 &= AC^2\\25+3025 &= AC^2\\3050 &= AC^2\\AC &= \sqrt{3050}=5\sqrt{122}

3. Because AE is tangent to both circles, it is perpendicular to both radii and \triangle ABC and \triangle DBE are similar. To find DB , use the Pythagorean Theorem.

10^2+24^2 &= DB^2\\100+576 &= 676\\DB &= \sqrt{676}=26

To find BC , use similar triangles.

\frac{5}{10}=\frac{BC}{26} & \longrightarrow BC=13\\DC=AB+BC &= 26+13=39

Interactive Practice

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

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

  1. A and B are points of tangency for \bigodot C and \bigodot D , respectively.
    1. Is \triangle AEC \sim \triangle BED ? Why?
    2. Find BC .
    3. Find AD .
    4. Using the trigonometric ratios, find m \angle C . Round to the nearest tenth of a degree.
  2. 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
2. \overline{AD} \cong \overline{DC}
3. \overline{DA} \bot \overline{AB} and \overline{DC} \bot \overline{CB}
4. Definition of perpendicular lines
5. Connecting two existing points
6. \triangle ADB and \triangle DCB are right triangles
7. \overline{DB} \cong \overline{DB}
8. \triangle ABD \cong \triangle CBD
9. \overline{AB} \cong \overline{CB}
  1. From the above proof, we can also conclude (fill in the blanks):
    1. ABCD is a _____________ (type of quadrilateral).
    2. The line that connects the ___________ and the external point B _________ \angle ADC and \angle ABC .
  2. Points A, B, C , and D are all points of tangency for the three tangent circles. Explain why \overline{AT} \cong \overline{BT} \cong \overline{CT} \cong \overline{DT} .
  3. Circles tangent at T are centered at M and N . \overline{ST} is tangent to both circles at T . Find the radius of the smaller circle if \overline{SN} \bot \overline{SM} , SM=22, TN=25 and m \angle SNT=40^\circ .
  4. Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to 16 cm, what is the radius of the bigger circle?
  5. Circles centered at A and B are tangent at W . Explain why A, B and W are collinear.




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

point of tangency

The point where the tangent line touches the circle.

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