Tangents to Circles10.1

Chapter 10Circles

Section 10.1

Tangents

Identify segments and lines related to circles

Use properties of tangents to circles.

Tangents to Circles10.1

Key Concepts

Example 1 Identifying Parts of a Circle

Example 2 Finding length of Radii

Example 3 Line Tangent to a Circle

Example 4 Determining Tangency

Example 5 Finding Lengths of Tangents

Identify segments and lines related to circles

Use properties of tangents to circles.

Tangents to Circles10.1

Circles

Definition

Definition Circle

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

C

Circle C

The distance from the center to a point on the

circle is called the radius

Tangents to Circles10.1

Circles

Any Segment with endpoints on the circle

are called chords

Any segment with endpoints being the center and any

point on the circle is called the Radius

A chord that goes through the center is called a diameter

A diameter is composed of two radii

Tangents to Circles10.1

Circles

Secant: A line in the same plane that

intersects a circle twice

Tangent: A line in the same plane that intersects

a circle exactly once

Point of Tangency

Tangents to Circles10.1

Circles

Tangents to Circles10.1

Name the circle.

Answer: The circle has its center at E, so it is named circle E, or .

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: Four radii are shown: .

Name the radius of the circle.

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: Four chords are shown: .

Name a chord of the circle.

Identify segments and lines related to circles

Tangents to Circles10.1

Name a diameter of the circle.

Answer: are the only chords that go through the center. So, are diameters.

Identify segments and lines related to circles

Tangents to Circles10.1

Answer:

Answer:

a. Name the circle.

b. Name a radius of the circle.

c. Name a chord of the circle.

d. Name a diameter of the circle.

Answer:

Answer:

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: 9

Formula for radius

Substitute and simplify.

If ST 18, find RS.

Circle R has diameters and .

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: 48

Formula for diameter

Substitute and simplify.

If RM 24, find QM.

Circle R has diameters .

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: So, RP = 2.

Since all radii are congruent, RN = RP.

If RN 2, find RP.

Circle R has diameters .

Identify segments and lines related to circles

Tangents to Circles10.1

Answer: 58

Answer: 12½

a. If BG = 25, find MG.

b. If DM = 29, find DN.

Circle M has diameters

c. If MF = 8.5, find MG.

Answer: 8½

Identify segments and lines related to circles

Tangents to Circles10.1

Use properties of tangents to circles.

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Tangents to Circles10.1

ALGEBRA is tangent to at point R. Find y.

Because the radius is perpendicular to the tangent at the point of tangency, . This makes a right angle and a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

Use properties of tangents to circles.

Tangents to Circles10.1

Pythagorean Theorem

Simplify.

Subtract 256 from each side.

Take the square root of each side.

Because y is the length of the diameter, ignore the negative result.

Answer: Thus, y is twice .

Use properties of tangents to circles.

Tangents to Circles10.1

Answer: 15

is a tangent to at point D. Find a.

Use properties of tangents to circles.

Tangents to Circles10.1

Use properties of tangents to circles.

If a line is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

Tangents to Circles10.1

First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

Use properties of tangents to circles.

Tangents to Circles10.1

Pythagorean Theorem

Simplify.

Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle.

Answer: So, is not tangent to .

Use properties of tangents to circles.

Tangents to Circles10.1

First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

Use properties of tangents to circles.

Tangents to Circles10.1

Pythagorean Theorem

Simplify.

Answer: Thus, making a tangent to

Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle.

Use properties of tangents to circles.

Tangents to Circles10.1

Answer: yes

a. Determine whether is tangent to

Use properties of tangents to circles.

Tangents to Circles10.1

Answer: no

b. Determine whether is tangent to

Use properties of tangents to circles.

Tangents to Circles10.1

Use properties of tangents to circles.

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Tangents to Circles10.1

ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent.

are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to

Use properties of tangents to circles.

Tangents to Circles10.1

Definition of congruent segments

Substitution.

Use the value of y to find x.

Definition of congruent segments

Substitution

Simplify.

Subtract 14 from each side.

Answer: 1

Use properties of tangents to circles.

Tangents to Circles10.1

ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent.

Answer: –6

Use properties of tangents to circles.

Tangents to Circles10.1

Triangle HJK is circumscribed about Find the perimeter of HJK if

Use properties of tangents to circles.

Tangents to Circles10.1

Use Theorem 10.3 to determine the equal measures.

We are given that

Answer: The perimeter of HJK is 158 units.

Definition of perimeter

Substitution

Use properties of tangents to circles.

Tangents to Circles10.1

Triangle NOT is circumscribed about Find the perimeter of NOT if

Answer: 172 units

Use properties of tangents to circles.

Tangents to Circles10.1

HW # 34Pg 599-600 9-35