Properties Properties of of

TangentsTangents

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.

Theorem 6.1

In a plane, a line segment 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.

EXAMPLE 3 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 6.1, ST is tangent to P.

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

EXAMPLE 4 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 6.1 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 6.2

Tangent segments from a common external point are

congruent.

EXAMPLE 5

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

Is DE tangent to C?

ANSWER

Yes

GUIDED PRACTICE

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

ANSWER

r = 7

GUIDED PRACTICE

Find the value(s) of x.

+3 = x

ANSWER