Tangency. Lines of Circles EXAMPLE 1 Identify special segments and lines Tell whether the line, ray, or segment is best described as a radius, chord,

Tangency

Lines of Circles

EXAMPLE 1 Identify special segments and lines

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

ACa.

SOLUTION

is a radius because C is the center and A is a point on the circle.

ACa.

EXAMPLE 1 Identify special segments and lines

b. AB is a diameter because it is a chord that contains the center C.

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

b. AB

SOLUTION

EXAMPLE 1 Identify special segments and lines

c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.

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

SOLUTION

DEc.

EXAMPLE 1 Identify special segments and lines

d. AE is a secant because it is a line that intersects the circle in two points.

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

SOLUTION

AEd.

SOLUTION

GUIDED PRACTICE for Example 1

Is a chord because it is a segment whose endpoints are on the circle.

AG

CB is a radius because C is the center and B is a point on the circle.

1. In Example 1, what word best describesAG ? CB ?

SOLUTION

GUIDED PRACTICE for Example 1

2. In Example 1, name a tangent and a tangent segment.

A tangent is DE

A tangent segment is DB

EXAMPLE 2 Find lengths in circles in a coordinate plane

b. Diameter of A

Radius of Bc.

Diameter of Bd.

Use the diagram to find the given lengths.

a. Radius of A

SOLUTION

a. The radius of A is 3 units.

b. The diameter of A is 6 units.

c. The radius of B is 2 units.

d. The diameter of B is 4 units.

SOLUTION

GUIDED PRACTICE for Example 2

a. The radius of C is 3 units.

b. The diameter of C is 6 units.

c. The radius of D is 2 units.

d. The diameter of D is 4 units.

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

Diagram Common tangents

EXAMPLE 3 Draw common tangents

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

a. b. c.

SOLUTION

a. 4 common tangents 3 common tangentsb.

EXAMPLE 3 Draw common tangents

c. 2 common tangents

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

c.

SOLUTION

SOLUTION

GUIDED PRACTICE for Example 3

Tell how many common tangents the circles have and draw them.4.

2 common tangents

SOLUTION

GUIDED PRACTICE for Example 3

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

1 common tangent

5.

SOLUTION

GUIDED PRACTICE for Example 3

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

No common tangents

6.

Theorem Tangent Perpendicular

Theorem Tangent Perpendicular Converse

EXAMPLE 4 Verify a tangent to a circle

SOLUTION

Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, 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.

In the diagram, PT is a radius of P. Is ST tangent to P ?

EXAMPLE 5 Find the radius of a circle

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

SOLUTION

You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem.

AC2 = BC2 + AB2

(r + 50)2 = r2 + 802

r2 + 100r + 2500 = r2 + 6400

100r = 3900

r = 39 ft .

Pythagorean Theorem

Substitute.

Multiply.

Subtract from each side.

Divide each side by 100.

Theorem Common Point Tangency

EXAMPLE 6 Find the radius of a circle

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

SOLUTION

RS = RT

28 = 3x + 4

8 = x

Substitute.

Solve for x.

Tangent segments from the same point are

GUIDED PRACTICE for Examples 4, 5 and 6

7. Is DE tangent to C?

ANSWER

Yes

GUIDED PRACTICE for Examples 4, 5 and 6

8. ST is tangent to Q.Find the value of r.

ANSWER

r = 7

GUIDED PRACTICE for Examples 4, 5 and 6

9. Find the value(s) of x.

+3 = x

ANSWER