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Tangent Lines

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

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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.

Watch This

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.

\begin{align*}\overleftrightarrow{BC}\end{align*} is tangent at point @$\begin{align*}B\end{align*}@$ if and only if @$\begin{align*}\overleftrightarrow{BC} \perp \overline{AB}\end{align*}@$.

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.

@$\begin{align*}\overline{BC}\end{align*}@$ and @$\begin{align*}\overline{DC}\end{align*}@$ have @$\begin{align*}C\end{align*}@$ as an endpoint and are tangent; @$\begin{align*}\overline{BC} \cong \overline{DC}\end{align*}@$.

Example A

@$\begin{align*}\overline{CB}\end{align*}@$ is tangent to @$\begin{align*}\bigodot A\end{align*}@$ at point @$\begin{align*}B\end{align*}@$. Find @$\begin{align*}AC\end{align*}@$. Reduce any radicals.

@$\begin{align*}\overline{CB}\end{align*}@$ is tangent, so @$\begin{align*}\overline{AB} \perp \overline{CB}\end{align*}@$ and @$\begin{align*}\triangle ABC\end{align*}@$ a right triangle. Use the Pythagorean Theorem to find @$\begin{align*}AC\end{align*}@$.

@$$\begin{align*}5^2+8^2&=AC^2\\ 25+64&=AC^2\\ 89&=AC^2\\ AC&=\sqrt{89}\end{align*}@$$

Example B

Using the answer from Example A above, find @$\begin{align*}DC\end{align*}@$ in @$\begin{align*}\bigodot A\end{align*}@$. Round your answer to the nearest hundredth.

@$$\begin{align*}DC & = AC - AD\\ DC & = \sqrt{89}-5 \approx 4.43\end{align*}@$$

Example C

Find the perimeter of @$\begin{align*}\triangle ABC\end{align*}@$.

@$\begin{align*}AE = AD, \ EB = BF,\end{align*}@$ and @$\begin{align*}CF = CD\end{align*}@$. Therefore, the perimeter of @$\begin{align*}\triangle ABC=6+6+4+4+7+7=34\end{align*}@$.

@$\begin{align*}\bigodot G\end{align*}@$ is inscribed in @$\begin{align*}\triangle ABC\end{align*}@$. A circle is inscribed in a polygon if every side of the polygon is tangent to the circle.

Tangent Lines CK-12

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Guided Practice

1. Find @$\begin{align*}AB\end{align*}@$ between @$\begin{align*}\bigodot A\end{align*}@$ and @$\begin{align*}\bigodot B\end{align*}@$. Reduce the radical.

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

3. If @$\begin{align*}D\end{align*}@$ and @$\begin{align*}C\end{align*}@$ are the centers and @$\begin{align*}AE\end{align*}@$ is tangent to both circles, find @$\begin{align*}DC\end{align*}@$.

4. Find the value of @$\begin{align*}x\end{align*}@$.

Answers:

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

@$$\begin{align*}5^2+55^2&=AB^2\\ 25+3025&=AB^2\\ 3050&=AB^2\\ AB&=\sqrt{3050}=5\sqrt{122}\end{align*}@$$

2. Again, use the Pythagorean Theorem. @$\begin{align*}4\sqrt{10}\end{align*}@$ is the longest side, so it will be @$\begin{align*}c\end{align*}@$.

Does @$$\begin{align*}8^2+10^2 \ & = \ \left ( 4\sqrt{10} \right )^2?\\ 64+100 & \ne 160\end{align*}@$$

@$\begin{align*}\triangle ABC\end{align*}@$ is not a right triangle. From this, we also find that @$\begin{align*}\overline{CB}\end{align*}@$ is not tangent to @$\begin{align*}\bigodot A\end{align*}@$.

3. @$\begin{align*}\overline{AE} \perp \overline{DE}\end{align*}@$ and @$\begin{align*}\overline{AE} \perp \overline{AC}\end{align*}@$ and @$\begin{align*}\triangle ABC \sim \triangle DBE\end{align*}@$ by AA Similarity.

To find @$\begin{align*}DB\end{align*}@$, use the Pythagorean Theorem.

@$$\begin{align*}10^2+24^2&=DB^2\\ 100+576&=676\\ DB&=\sqrt{676}=26\end{align*}@$$

To find @$\begin{align*}BC\end{align*}@$, use similar triangles. @$\begin{align*}\frac{5}{10}=\frac{BC}{26} \longrightarrow BC=13. \ DC=DB+BC=26+13=39\end{align*}@$

4. @$\begin{align*}\overline{AB} \cong \overline{CB}\end{align*}@$ by the Two Tangents Theorem. Set @$\begin{align*}AB = CB\end{align*}@$ and solve for @$\begin{align*}x\end{align*}@$.

@$$\begin{align*}4x-9&=15\\ 4x&=24\\ x&=6\end{align*}@$$

Explore More

Determine whether the given segment is tangent to @$\begin{align*}\bigodot K\end{align*}@$.

Find the value of the indicated length(s) in @$\begin{align*}\bigodot C\end{align*}@$. @$\begin{align*}A\end{align*}@$ and @$\begin{align*}B\end{align*}@$ are points of tangency. Simplify all radicals.

@$\begin{align*}A\end{align*}@$ and @$\begin{align*}B\end{align*}@$ are points of tangency for @$\begin{align*}\bigodot C\end{align*}@$ and @$\begin{align*}\bigodot D\end{align*}@$.

  1. Is @$\begin{align*}\triangle AEC \sim \triangle BED\end{align*}@$? Why?
  2. Find @$\begin{align*}CE\end{align*}@$.
  3. Find @$\begin{align*}BE\end{align*}@$.
  4. Find @$\begin{align*}ED\end{align*}@$.
  5. Find @$\begin{align*}BC\end{align*}@$ and @$\begin{align*}AD\end{align*}@$.

@$\begin{align*}\bigodot A\end{align*}@$ is inscribed in @$\begin{align*}BDFH\end{align*}@$.

  1. Find the perimeter of @$\begin{align*}BDFH\end{align*}@$.
  2. What type of quadrilateral is @$\begin{align*}BDFH\end{align*}@$? 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: @$\begin{align*}\overline{AB}\end{align*}@$ and @$\begin{align*}\overline{CB}\end{align*}@$ with points of tangency at @$\begin{align*}A\end{align*}@$ and @$\begin{align*}C\end{align*}@$. @$\begin{align*}\overline{AD}\end{align*}@$ and @$\begin{align*}\overline{DC}\end{align*}@$ are radii.

Prove: @$\begin{align*}\overline{AB} \cong \overline{CB}\end{align*}@$

Statement Reason
1. 1.
2. @$\begin{align*}\overline{AD} \cong \overline{DC}\end{align*}@$ 2.
3. @$\begin{align*}\overline{DA} \perp \overline{AB}\end{align*}@$ and @$\begin{align*}\overline{DC} \perp \overline{CB}\end{align*}@$ 3.
4. 4. Definition of perpendicular lines
5. 5. Connecting two existing points
6. @$\begin{align*}\triangle ADB\end{align*}@$ and @$\begin{align*}\triangle DCB\end{align*}@$ are right triangles 6.
7. @$\begin{align*}\overline{DB} \cong \overline{DB}\end{align*}@$ 7.
8. @$\begin{align*}\triangle ABD \cong \triangle CBD\end{align*}@$ 8.
9. @$\begin{align*}\overline{AB} \cong \overline{CB}\end{align*}@$ 9.
  1. Fill in the blanks, using the proof from #21.
    1. @$\begin{align*}ABCD\end{align*}@$ is a _____________ (type of quadrilateral).
    2. The line that connects the ___________ and the external point @$\begin{align*}B\end{align*}@$ __________ @$\begin{align*}\angle ABC\end{align*}@$.
  2. Points @$\begin{align*}A, \ B,\end{align*}@$ and @$\begin{align*}C\end{align*}@$ are points of tangency for the three tangent circles. Explain why @$\begin{align*}\overline{AT} \cong \overline{BT} \cong \overline{CT}\end{align*}@$.

Vocabulary

circle

circle

The set of all points that are the same distance away from a specific point, called the center.
diameter

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

point of tangency

The point where the tangent line touches the circle.
radius

radius

The distance from the center to the outer rim of a circle.
Tangent

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

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

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.

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