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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

First watch this video.

Now watch this video.

### 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.

### 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 $x$ .

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 $\overline{AT} \cong \overline{BT} \cong \overline{CT}$ .

### 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.