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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.

BC\begin{align*}\overleftrightarrow{BC}\end{align*} is tangent at point B\begin{align*}B\end{align*} if and only if BCAB¯¯¯¯¯\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.

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

#### Example A

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

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

52+8225+6489AC=AC2=AC2=AC2=89

#### Example B

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

#### Example C

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

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

G\begin{align*}\bigodot G\end{align*} is inscribed in ABC\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 AB\begin{align*}AB\end{align*} between A\begin{align*}\bigodot A\end{align*} and B\begin{align*}\bigodot B\end{align*}. Reduce the radical.

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

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

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

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

52+55225+30253050AB=AB2=AB2=AB2=3050=5122

2. Again, use the Pythagorean Theorem. 410\begin{align*}4\sqrt{10}\end{align*} is the longest side, so it will be c\begin{align*}c\end{align*}.

Does

82+102 64+100= (410)2?160

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

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

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

102+242100+576DB=DB2=676=676=26

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*}.

### 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*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.2.

### Vocabulary Language: English Spanish

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.

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.