## Key Concepts

- Draw tangents to a circle
- Identify special segments and lines.
- Find lengths in a coordinate plane.
- Draw common tangents.
- Verify a tangent to a circle.
- Find the radius of a circle.
- Tangents from a common external point are congruent.

## Tangents

**Question: **

How are the lengths of tangent segments related?

A line can intersect a circle at 0, 1, or 2 points. If a line is in the plane of a circle and intersects the circle at 1 point, the line is a tangent.

**Step 1:**

Draw tangents to a circle

Draw a circle. Use a compass to draw a circle. Label the center P.

Draw tangents

Draw lines and BC so that they intersect ⊙ P only at A and C, respectively. These lines are called tangents.

Measure segments

AB↔ and BC↔ are called tangent segments. Measure and compare the lengths of the tangent segments.

A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.

A circle with center P is called “circle P” and can be written ⊙ P.

A **segment** whose endpoints are the center and any point on the circle is a **radius**.

A** chord **is a segment whose endpoints are on a circle.

A **diameter** is a chord that contains the center of the circle.

A **secant** is a line that intersects a circle in two points.

A **tangent** is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.

The **tangent** ray

AB and the tangent **segment** AB are also called **tangents**.

**Example 1:**

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ⊙C.

a. AC

b. AB

c. DE

d. AE

**Solution **

- AC is a radius because C is the center and A is a point on the circle.
- AB is a diameter because it is a chord that contains the center C.
- DE is a tangent ray because it is contained in a line that intersects the circle at only one point.
- AE is a secant because it is a line that intersects the circle in two points.

### Guided Practice

**Example 1**:

- In Example 1, what word best describes AG and CB ?
- In Example 1, name a tangent and a tangent segment.

**Solution:**

- AG = Chord/Diameter, CB =radius,
- DE = Tangent and DE

### Read Vocabulary

The plural of radius is radii. All radii of a circle are congruent.

### Radius and Diameter

The words radius and diameter are used for lengths as well as segments.

For a given circle, think of a radius and a diameter as segments and the radius and the diameter as lengths.

### Find lengths in circles in a coordinate plane

Use the diagram to find the given lengths.

- Radius of A
- Diameter of A
- Radius of B
- Diameter of B

Solution: Count squares

- The radius of ⨀A is 3 units.
- The diameter of ⨀A is 6 units.
- The radius of ⨀B is 2 units.
- The diameter of ⨀B is 4 units.

### Guided Practice

**Example 2: **

Use the diagram in Example 2 to find the radius and diameter of ⨀C and ⨀D.

**Solution:**

The radius of ⨀C is 3 units.

The diameter of ⨀C is 6 units

The radius of ⨀D is 2 units.

The radius of ⨀D is 4 units.

### Coplanar Circles

Two circles can intersect in two points, one point, or no points.

Coplanar circles that intersect in one point are called tangent circles.

Coplanar circles that have a common center are called concentric circles.

### Common Tangents

A line, ray, or segment that is tangent to two coplanar circles is called a common tangent.

**Draw common tangents**** **

### Guided Practice

**Example 3**:

Tell how many common tangents the circles have and draw them.

### Theorem 10.1

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

GIVEN: Line *m* is tangent to ⨀Q at P.

PROVE c: m ⊥ QP

**1. Assume m is not perpendicular to QP**

Then the perpendicular segment from Q to *m* intersects *m* at some other point R.

Because *m* is a tangent, R cannot be inside ⨀Q. Compare the length QR to QP.

- Because
**QR**is the perpendicular segment from Q to*m*. QR is the shortest segment from Q to*m*. Now compare QR to QP.

- Use your results from parts (a) and (b) to complete the indirect proof.

Verify a tangent to a circle

**EXAMPLE 4**:

In the diagram

PT is a radius of ⊙P. Is ST tangent to ⨀P?

**Solution:**

Use the Converse of the Pythagorean Theorem.

Because 12^{2 }+ 35^{2}= 37^{2} △PST is a right triangle and ST ⊥ PT

So, ST is perpendicular to a radius of ⨀P at its endpoint on ⨀P.

By Theorem 10.1,

ST is tangent to ⨀P.

**EXAMPLE 5:**

In the diagram, B is a point of tangency. Find the radius* r* of ⨀C.

You know from Theorem 10.1 that

AB ⊥ BC , so △ABC is a right triangle.

You can use the Pythagorean Theorem.

AC^{2 }=BC ^{2} +AB ^{2 } (Pythagorean Theorem)

(r + 50)^{2} = r ^{2} + 80^{2} (Substitute)

r ^{2} + 100r + 2500 = r ^{2} + 6400 (Multiply)

100r = 3900 (Subtract from each side)

∴ r =39 ft (Divide each side by 100)

### Theorem 10.2

Tangent segments from a common external point are congruent.

**Proving Theorem10.2**

Write proof that tangent segments from a common external point are congruent.

GIVEN:

SR and ST are tangent to ⊙P.

PROVE: SR > ST

**Plan for Proof **

Use the hypotenuse–leg congruence

Theorem to show that

△ SRP △STP

Join P, A; PB and PS

**∠PRS=** **∠STP=** 900

Angle between radii and tangents

Now, the right triangles

∆PRS and ∆PTS

PR=PT (radii of the same circle)

PS=PS (Common)

Therefore, by R.H.S congruency axiom

∆PRS ≅ ∆PTS

This gives SR ≅ ST

**Hence proved. **

**Example 6**:

RS is tangent to ⨀C at S and RT is tangent to ⨀C at T. Find the value of x.

**Solution:**

RT = RS (Tangent segments from the same point are ≅)

3x+4 =28 (Substitute)

3x =28-4

3x=24

x=8 (solve for *x)*

### Problem solving

#### GLOBAL POSITIONING SYSTEM (GPS)

GPS satellites orbit about 11,000 miles above Earth. The mean radius of Earth is about 3959 miles. Because GPS signals cannot travel through Earth, a satellite can transmit signals only as far as points A and C from point B, as shown. Find BA and BC to the nearest mile.

**Solution:**

ED =3959 mi is the radius of Earth

DB=1100 mi

AB=? BC=? AB, BC are tangents

AB=BC (∵ Tangent segments from a common external point are congruent.)

AE=ED Radius of Earth

BE=ED+DB=3959+11000=14959 mi

AE ⊥ AB The tangent at any point of a circle is perpendicular to the radius through the point of contact.

△EAB is right angled triangle. By Pythagoras theorem

AE^{2 }+AB^{2}= BE^{2}

AB^{2} = BE^{2}– AE^{2}

= 14959^{2} – 3959^{2}

= (14959+3959) (14959–3959)

=11000 x 18918

AB= √18918 × 11000

=14425.60 mi

### Let’s check your knowledge

- Copy the diagram. Tell how many common tangents the circles have and draw.

- Determine whether AB is to ⊙C. Explain.

- Find the values of the variable.

- Find the values of
*x*and*y.*

### Answers

2. From △CAB

BC^{2} = AC^{2}+ AB^{2} (∵ Pythagoras theorem)

5^{2} = 3^{2}+ 4^{2 }

25=25

So△CAB is right angled triangle.

∠CAB = 900

CA ⊥ AB

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ AB is a tangent

3. Tangent segments from a common external point are congruent.

So, AD=AB

7x – 6 =3x+10

7x-3x=10+6

4x=16

∴ x=4

**Step 1:**

Linear pair = 180^{0}

(2x+3)^{0 }+ 137^{0}= 180^{0}

x^{0} + 140^{0}= 180^{0}

2x^{0} = 40^{0}

x^{0} = 20^{0}

(2x+3)^{0} =2 x 20^{0}+3= 43^{0}

**Step 2:** Two lines are parallel to each other, then

Alternate exterior angles are equal

(4y−7)^{o} = (2x+3)^{0}

(4y−7)^{o} = 43^{0}

4y^{0} = 50^{0}

y^{0} = (252)^{0}

## Exercise

- Find the value of x and y in the following figure

- When will two lines tangent to the same circle not intersect?
- Find the value of the variables in the following figures:

- Determine whether AB tangent is tangent to circle C. Explain.

### Concept map

### What have we learned

- How to draw tangents to a circle.
- How to Identify special segments and lines.
- How to find lengths in a coordinate plane.
- How to draw common tangents.
- How to verify a tangent to a circle.
- How to find the radius of a circle.
- How tangents from a common external point are congruent

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