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Tangent Lines

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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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Tangent Lines CK-12

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.

\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.

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

Example A

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

\overline{CB} is tangent, so \overline{AB} \perp \overline{CB} and \triangle ABC a right triangle. 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

Using the answer from Example A above, 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 .

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

\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.

Tangent Lines CK-12

Guided Practice

1. Find AB between \bigodot A and \bigodot B . Reduce the radical.

2. Determine if the triangle below is a right triangle.

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

4. Find the value of x .

Answers:

1. \overline{AD} \perp \overline{DC} and \overline{DC} \perp \overline{CB} . Draw in \overline{BE} , so EDCB is a rectangle. Use the Pythagorean Theorem to find AB .

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

2. Again, use the Pythagorean Theorem. 4\sqrt{10} is the longest side, so it will be c .

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

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

3. \overline{AE} \perp \overline{DE} and \overline{AE} \perp \overline{AC} and \triangle ABC \sim \triangle DBE by AA Similarity.

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=DB+BC=26+13=39

4. \overline{AB} \cong \overline{CB} by the Two Tangents Theorem. Set AB = CB and solve for x .

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

Practice

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

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

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

  1. Is \triangle AEC \sim \triangle BED ? Why?
  2. Find CE .
  3. Find BE .
  4. Find ED .
  5. Find BC and AD .

\bigodot A is inscribed in BDFH .

  1. Find the perimeter of BDFH .
  2. What type of quadrilateral is BDFH ? How do you know?
  3. Draw a circle inscribed in a square. If the radius of the circle is 5, what is the perimeter of the square?
  4. Can a circle be inscribed in a rectangle? If so, draw it. If not, explain.
  5. Draw a triangle with two sides tangent to a circle, but the third side is not.
  6. Can a circle be inscribed in an obtuse triangle? If so, draw it. If not, explain.
  7. 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. \triangle ADB and \triangle DCB are right triangles 6.
7. \overline{DB} \cong \overline{DB} 7.
8. \triangle ABD \cong \triangle CBD 8.
9. \overline{AB} \cong \overline{CB} 9.
  1. 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 .
  2. Points A, \ B, and C are points of tangency for the three tangent circles. Explain why \overline{AT} \cong \overline{BT} \cong \overline{CT} .

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